TheInfoList

OR:

In
ring theory In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their rep ...
, a branch of
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
, an ideal of a ring is a special
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
of its elements. Ideals generalize certain subsets of the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
s, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer (even or odd) results in an even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring in a way similar to how, in
group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...
, a
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
can be used to construct a quotient group. Among the integers, the ideals correspond one-for-one with the
non-negative integer In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal n ...
s: in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number. However, in other rings, the ideals may not correspond directly to the ring elements, and certain properties of integers, when generalized to rings, attach more naturally to the ideals than to the elements of the ring. For instance, the
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together wi ...
s of a ring are analogous to
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s, and the
Chinese remainder theorem In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of the ...
can be generalized to ideals. There is a version of unique prime factorization for the ideals of a Dedekind domain (a type of ring important in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathe ...
). The related, but distinct, concept of an ideal in
order theory Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...
is derived from the notion of ideal in ring theory. A fractional ideal is a generalization of an ideal, and the usual ideals are sometimes called integral ideals for clarity.

# History

Ernst Kummer invented the concept of ideal numbers to serve as the "missing" factors in number rings in which unique factorization fails; here the word "ideal" is in the sense of existing in imagination only, in analogy with "ideal" objects in geometry such as points at infinity. In 1876, Richard Dedekind replaced Kummer's undefined concept by concrete sets of numbers, sets that he called ideals, in the third edition of Dirichlet's book '' Vorlesungen über Zahlentheorie'', to which Dedekind had added many supplements. Later the notion was extended beyond number rings to the setting of polynomial rings and other commutative rings by
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
and especially Emmy Noether.

# Definitions and motivation

For an arbitrary ring $\left(R,+,\cdot\right)$, let $\left(R,+\right)$ be its additive group. A subset $I$ is called a left ideal of $R$ if it is an additive subgroup of $R$ that "absorbs multiplication from the left by elements of $R$"; that is, $I$ is a left ideal if it satisfies the following two conditions: # $\left(I,+\right)$ is a
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
of $\left(R,+\right),$ # For every $r \in R$ and every $x \in I$, the product $r x$ is in $I$. A right ideal is defined with the condition replaced by . A two-sided ideal is a left ideal that is also a right ideal, and is sometimes simply called an ideal. In the language of modules, the definitions mean that a left (resp. right, two-sided) ideal of ''R'' is an ''R''-
submodule In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the ...
of ''R'' when ''R'' is viewed as a left (resp. right, bi-) ''R''-module. When ''R'' is a commutative ring, the definitions of left, right, and two-sided ideal coincide, and the term ideal is used alone. To understand the concept of an ideal, consider how ideals arise in the construction of rings of "elements modulo". For concreteness, let us look at the ring ℤ/''n''ℤ of integers modulo ''n'' given integer (note that ℤ is a commutative ring). The key observation here is that we obtain ℤ/''n''ℤ by taking the integer line ℤ and wrapping it around itself so that various integers get identified. In doing so, we must satisfy 2 requirements: 1) ''n'' must be identified with 0 since ''n'' is congruent to 0 modulo ''n.'' 2) the resulting structure must again be a ring. The second requirement forces us to make additional identifications (i.e., it determines the precise way in which we must wrap ℤ around itself). The notion of an ideal arises when we ask the question:
What is the exact set of integers that we are forced to identify with 0?
The answer is, unsurprisingly, the set of all integers congruent to 0 modulo ''n''. That is, we must wrap ℤ around itself infinitely many times so that the integers ..., , , , , ... will all align with 0. If we look at what properties this set must satisfy in order to ensure that ℤ/''n''ℤ is a ring, then we arrive at the definition of an ideal. Indeed, one can directly verify that ''n''ℤ is an ideal of ℤ. Remark. Identifications with elements other than 0 also need to be made. For example, the elements in must be identified with 1, the elements in must be identified with 2, and so on. Those, however, are uniquely determined by ''n''ℤ since ℤ is an additive group. We can make a similar construction in any commutative ring ''R'': start with an arbitrary , and then identify with 0 all elements of the ideal It turns out that the ideal ''xR'' is the smallest ideal that contains ''x'', called the ideal generated by ''x''. More generally, we can start with an arbitrary subset , and then identify with 0 all the elements in the ideal generated by ''S'': the smallest ideal (''S'') such that . The ring that we obtain after the identification depends only on the ideal (''S'') and not on the set ''S'' that we started with. That is, if , then the resulting rings will be the same. Therefore, an ideal ''I'' of a commutative ring ''R'' captures canonically the information needed to obtain the ring of elements of ''R'' modulo a given subset . The elements of ''I'', by definition, are those that are congruent to zero, that is, identified with zero in the resulting ring. The resulting ring is called the
quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
of ''R'' by ''I'' and is denoted ''R''/''I''. Intuitively, the definition of an ideal postulates two natural conditions necessary for ''I'' to contain all elements designated as "zeros" by ''R''/''I'': # ''I'' is an additive subgroup of ''R'': the zero 0 of ''R'' is a "zero" , and if and are "zeros", then is a "zero" too. # Any multiplied by a "zero" is a "zero" . It turns out that the above conditions are also sufficient for ''I'' to contain all the necessary "zeros": no other elements have to be designated as "zero" in order to form ''R''/''I''. (In fact, no other elements should be designated as "zero" if we want to make the fewest identifications.) Remark. The above construction still works using two-sided ideals even if ''R'' is not necessarily commutative.

# Examples and properties

(For the sake of brevity, some results are stated only for left ideals but are usually also true for right ideals with appropriate notation changes.) * In a ring ''R'', the set ''R'' itself forms a two-sided ideal of ''R'' called the unit ideal. It is often also denoted by $\left(1\right)$ since it is precisely the two-sided ideal generated (see below) by the unity $1_R$. Also, the set $\$ consisting of only the additive identity 0''R'' forms a two-sided ideal called the zero ideal and is denoted by $\left(0\right)$.Some authors call the zero and unit ideals of a ring ''R'' the trivial ideals of ''R''. Every (left, right or two-sided) ideal contains the zero ideal and is contained in the unit ideal. * An (left, right or two-sided) ideal that is not the unit ideal is called a proper ideal (as it is a
proper subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
). Note: a left ideal $\mathfrak$ is proper if and only if it does not contain a unit element, since if $u \in \mathfrak$ is a unit element, then $r = \left(r u^\right) u \in \mathfrak$ for every $r \in R$. Typically there are plenty of proper ideals. In fact, if ''R'' is a skew-field, then $\left(0\right), \left(1\right)$ are its only ideals and conversely: that is, a nonzero ring ''R'' is a skew-field if $\left(0\right), \left(1\right)$ are the only left (or right) ideals. (Proof: if $x$ is a nonzero element, then the principal left ideal $Rx$ (see below) is nonzero and thus $Rx = \left(1\right)$; i.e., $yx = 1$ for some nonzero $y$. Likewise, $zy = 1$ for some nonzero $z$. Then $z = z\left(yx\right) = \left(zy\right)x = x$.) * The even
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
s form an ideal in the ring $\mathbb$ of all integers; it is usually denoted by $2\mathbb$. This is because the sum of any even integers is even, and the product of any integer with an even integer is also even. Similarly, the set of all integers divisible by a fixed integer ''n'' is an ideal denoted $n\mathbb$. * The set of all
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exam ...
s with real coefficients which are divisible by the polynomial ''x''2 + 1 is an ideal in the ring of all polynomials. * The set of all ''n''-by-''n'' matrices whose last row is zero forms a right ideal in the ring of all ''n''-by-''n'' matrices. It is not a left ideal. The set of all ''n''-by-''n'' matrices whose last ''column'' is zero forms a left ideal but not a right ideal. * The ring $C\left(\mathbb\right)$ of all
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in va ...
s ''f'' from $\mathbb$ to $\mathbb$ under pointwise multiplication contains the ideal of all continuous functions ''f'' such that ''f''(1) = 0. Another ideal in $C\left(\mathbb\right)$ is given by those functions which vanish for large enough arguments, i.e. those continuous functions ''f'' for which there exists a number ''L'' > 0 such that ''f''(''x'') = 0 whenever > ''L''. * A ring is called a simple ring if it is nonzero and has no two-sided ideals other than $\left(0\right), \left(1\right)$. Thus, a skew-field is simple and a simple commutative ring is a field. The matrix ring over a skew-field is a simple ring. * If $f: R \to S$ is a ring homomorphism, then the kernel $\ker\left(f\right) = f^\left(0_S\right)$ is a two-sided ideal of $R$. By definition, $f\left(1_R\right) = 1_S$, and thus if $S$ is not the zero ring (so $1_S\ne0_S$), then $\ker\left(f\right)$ is a proper ideal. More generally, for each left ideal ''I'' of ''S'', the pre-image $f^\left(I\right)$ is a left ideal. If ''I'' is a left ideal of ''R'', then $f\left(I\right)$ is a left ideal of the subring $f\left(R\right)$ of ''S'': unless ''f'' is surjective, $f\left(I\right)$ need not be an ideal of ''S''; see also #Extension and contraction of an ideal below. * Ideal correspondence: Given a surjective ring homomorphism $f: R \to S$, there is a bijective order-preserving correspondence between the left (resp. right, two-sided) ideals of $R$ containing the kernel of $f$ and the left (resp. right, two-sided) ideals of $S$: the correspondence is given by $I \mapsto f\left(I\right)$ and the pre-image $J \mapsto f^\left(J\right)$. Moreover, for commutative rings, this bijective correspondence restricts to prime ideals, maximal ideals, and radical ideals (see the Types of ideals section for the definitions of these ideals). * (For those who know modules) If ''M'' is a left ''R''-module and $S \subset M$ a subset, then the annihilator $\operatorname_R\left(S\right) = \$ of ''S'' is a left ideal. Given ideals $\mathfrak, \mathfrak$ of a commutative ring ''R'', the ''R''-annihilator of $\left(\mathfrak + \mathfrak\right)/\mathfrak$ is an ideal of ''R'' called the ideal quotient of $\mathfrak$ by $\mathfrak$ and is denoted by $\left(\mathfrak : \mathfrak\right)$; it is an instance of idealizer in commutative algebra. * Let $\mathfrak_i, i \in S$ be an ascending chain of left ideals in a ring ''R''; i.e., $S$ is a totally ordered set and $\mathfrak_i \subset \mathfrak_j$ for each $i < j$. Then the union $\textstyle \bigcup_ \mathfrak_i$ is a left ideal of ''R''. (Note: this fact remains true even if ''R'' is without the unity 1.) * The above fact together with Zorn's lemma proves the following: if $E \subset R$ is a possibly empty subset and $\mathfrak_0 \subset R$ is a left ideal that is disjoint from ''E'', then there is an ideal that is maximal among the ideals containing $\mathfrak_0$ and disjoint from ''E''. (Again this is still valid if the ring ''R'' lacks the unity 1.) When $R \ne 0$, taking $\mathfrak_0 = \left(0\right)$ and $E = \$, in particular, there exists a left ideal that is maximal among proper left ideals (often simply called a maximal left ideal); see Krull's theorem for more. *An arbitrary union of ideals need not be an ideal, but the following is still true: given a possibly empty subset ''X'' of ''R'', there is the smallest left ideal containing ''X'', called the left ideal generated by ''X'' and is denoted by $RX$. Such an ideal exists since it is the intersection of all left ideals containing ''X''. Equivalently, $RX$ is the set of all the (finite) left ''R''-linear combinations of elements of ''X'' over ''R'': *:$RX = \.$ :(since such a span is the smallest left ideal containing ''X''.)If ''R'' does not have a unit, then the internal descriptions above must be modified slightly. In addition to the finite sums of products of things in ''X'' with things in ''R'', we must allow the addition of ''n''-fold sums of the form , and ''n''-fold sums of the form for every ''x'' in ''X'' and every ''n'' in the natural numbers. When ''R'' has a unit, this extra requirement becomes superfluous. A right (resp. two-sided) ideal generated by ''X'' is defined in the similar way. For "two-sided", one has to use linear combinations from both sides; i.e., ::$RXR = \.\,$ *A left (resp. right, two-sided) ideal generated by a single element ''x'' is called the principal left (resp. right, two-sided) ideal generated by ''x'' and is denoted by $Rx$ (resp. $xR, RxR$). The principal two-sided ideal $RxR$ is often also denoted by $\left(x\right)$. If $X = \$ is a finite set, then $RXR$ is also written as $\left(x_1, \dots, x_n\right)$. * In the ring $\mathbb$ of integers, every ideal can be generated by a single number (so $\mathbb$ is a
principal ideal domain In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are princip ...
), as a consequence of Euclidean division (or some other way). *There is a bijective correspondence between ideals and
congruence relation In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done ...
s (equivalence relations that respect the ring structure) on the ring: Given an ideal ''I'' of a ring ''R'', let if . Then ~ is a congruence relation on ''R''. Conversely, given a congruence relation ~ on ''R'', let . Then ''I'' is an ideal of ''R''.

# Types of ideals

''To simplify the description all rings are assumed to be commutative. The non-commutative case is discussed in detail in the respective articles.'' Ideals are important because they appear as kernels of ring homomorphisms and allow one to define factor rings. Different types of ideals are studied because they can be used to construct different types of factor rings. * Maximal ideal: A proper ideal ''I'' is called a maximal ideal if there exists no other proper ideal ''J'' with ''I'' a proper subset of ''J''. The factor ring of a maximal ideal is a simple ring in general and is a field for commutative rings. * Minimal ideal: A nonzero ideal is called minimal if it contains no other nonzero ideal. *
Prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together wi ...
: A proper ideal ''I'' is called a prime ideal if for any ''a'' and ''b'' in ''R'', if ''ab'' is in ''I'', then at least one of ''a'' and ''b'' is in ''I''. The factor ring of a prime ideal is a prime ring in general and is an integral domain for commutative rings. * Radical ideal or semiprime ideal: A proper ideal ''I'' is called radical or semiprime if for any ''a'' in ''R'', if ''a''''n'' is in ''I'' for some ''n'', then ''a'' is in ''I''. The factor ring of a radical ideal is a semiprime ring for general rings, and is a reduced ring for commutative rings. *
Primary ideal In mathematics, specifically commutative algebra, a proper ideal ''Q'' of a commutative ring ''A'' is said to be primary if whenever ''xy'' is an element of ''Q'' then ''x'' or ''y'n'' is also an element of ''Q'', for some ''n'' > 0. Fo ...
: An ideal ''I'' is called a primary ideal if for all ''a'' and ''b'' in ''R'', if ''ab'' is in ''I'', then at least one of ''a'' and ''b''''n'' is in ''I'' for some
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal n ...
''n''. Every prime ideal is primary, but not conversely. A semiprime primary ideal is prime. * Principal ideal: An ideal generated by ''one'' element. * Finitely generated ideal: This type of ideal is finitely generated as a module. * Primitive ideal: A left primitive ideal is the annihilator of a simple left module. * Irreducible ideal: An ideal is said to be irreducible if it cannot be written as an intersection of ideals which properly contain it. * Comaximal ideals: Two ideals $\mathfrak, \mathfrak$ are said to be comaximal if $x + y = 1$ for some $x \in \mathfrak$ and $y \in \mathfrak$. * Regular ideal: This term has multiple uses. See the article for a list. * Nil ideal: An ideal is a nil ideal if each of its elements is nilpotent. * Nilpotent ideal: Some power of it is zero. * Parameter ideal: an ideal generated by a system of parameters. Two other important terms using "ideal" are not always ideals of their ring. See their respective articles for details: * Fractional ideal: This is usually defined when ''R'' is a commutative domain with quotient field ''K''. Despite their names, fractional ideals are ''R'' submodules of ''K'' with a special property. If the fractional ideal is contained entirely in ''R'', then it is truly an ideal of ''R''. * Invertible ideal: Usually an invertible ideal ''A'' is defined as a fractional ideal for which there is another fractional ideal ''B'' such that . Some authors may also apply "invertible ideal" to ordinary ring ideals ''A'' and ''B'' with in rings other than domains.

# Ideal operations

The sum and product of ideals are defined as follows. For $\mathfrak$ and $\mathfrak$, left (resp. right) ideals of a ring ''R'', their sum is :$\mathfrak+\mathfrak:=\$, which is a left (resp. right) ideal, and, if $\mathfrak, \mathfrak$ are two-sided, :$\mathfrak \mathfrak:=\,$ i.e. the product is the ideal generated by all products of the form ''ab'' with ''a'' in $\mathfrak$ and ''b'' in $\mathfrak$. Note $\mathfrak + \mathfrak$ is the smallest left (resp. right) ideal containing both $\mathfrak$ and $\mathfrak$ (or the union $\mathfrak \cup \mathfrak$), while the product $\mathfrak\mathfrak$ is contained in the intersection of $\mathfrak$ and $\mathfrak$. The distributive law holds for two-sided ideals $\mathfrak, \mathfrak, \mathfrak$, *$\mathfrak\left(\mathfrak + \mathfrak\right) = \mathfrak \mathfrak + \mathfrak \mathfrak$, *$\left(\mathfrak + \mathfrak\right) \mathfrak = \mathfrak\mathfrak + \mathfrak\mathfrak$. If a product is replaced by an intersection, a partial distributive law holds: :$\mathfrak \cap \left(\mathfrak + \mathfrak\right) \supset \mathfrak \cap \mathfrak + \mathfrak \cap \mathfrak$ where the equality holds if $\mathfrak$ contains $\mathfrak$ or $\mathfrak$. Remark: The sum and the intersection of ideals is again an ideal; with these two operations as join and meet, the set of all ideals of a given ring forms a
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies ...
modular lattice. The lattice is not, in general, a distributive lattice. The three operations of intersection, sum (or join), and product make the set of ideals of a commutative ring into a quantale. If $\mathfrak, \mathfrak$ are ideals of a commutative ring ''R'', then $\mathfrak \cap \mathfrak = \mathfrak \mathfrak$ in the following two cases (at least) *$\mathfrak + \mathfrak = \left(1\right)$ *$\mathfrak$ is generated by elements that form a regular sequence modulo $\mathfrak$. (More generally, the difference between a product and an intersection of ideals is measured by the
Tor functor In mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central concepts of homological algebra, in which ideas from algebraic topology are used to cons ...
: $\operatorname^R_1\left(R/\mathfrak, R/\mathfrak\right) = \left(\mathfrak \cap \mathfrak\right)/ \mathfrak \mathfrak.$) An integral domain is called a Dedekind domain if for each pair of ideals $\mathfrak \subset \mathfrak$, there is an ideal $\mathfrak$ such that $\mathfrak \mathfrak = \mathfrak \mathfrak$. It can then be shown that every nonzero ideal of a Dedekind domain can be uniquely written as a product of maximal ideals, a generalization of the
fundamental theorem of arithmetic In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the ord ...
.

# Examples of ideal operations

In $\mathbb$ we have :$\left(n\right)\cap\left(m\right) = \operatorname\left(n,m\right)\mathbb$ since $\left(n\right)\cap\left(m\right)$ is the set of integers which are divisible by both $n$ and $m$. Let

Ideals appear naturally in the study of modules, especially in the form of a radical. :''For simplicity, we work with commutative rings but, with some changes, the results are also true for non-commutative rings.'' Let ''R'' be a commutative ring. By definition, a primitive ideal of ''R'' is the annihilator of a (nonzero) simple ''R''-module. The Jacobson radical $J = \operatorname\left(R\right)$ of ''R'' is the intersection of all primitive ideals. Equivalently, :$J = \bigcap_ \mathfrak.$ Indeed, if $M$ is a simple module and ''x'' is a nonzero element in ''M'', then $Rx = M$ and $R/\operatorname\left(M\right) = R/\operatorname\left(x\right) \simeq M$, meaning $\operatorname\left(M\right)$ is a maximal ideal. Conversely, if $\mathfrak$ is a maximal ideal, then $\mathfrak$ is the annihilator of the simple ''R''-module $R/\mathfrak$. There is also another characterization (the proof is not hard): :$J = \.$ For a not-necessarily-commutative ring, it is a general fact that $1 - yx$ is a unit element if and only if $1 - xy$ is (see the link) and so this last characterization shows that the radical can be defined both in terms of left and right primitive ideals. The following simple but important fact ( Nakayama's lemma) is built-in to the definition of a Jacobson radical: if ''M'' is a module such that $JM = M$, then ''M'' does not admit a maximal submodule, since if there is a maximal submodule $L \subsetneq M$, $J \cdot \left(M/L\right) = 0$ and so $M = JM \subset L \subsetneq M$, a contradiction. Since a nonzero finitely generated module admits a maximal submodule, in particular, one has: :If $JM = M$ and ''M'' is finitely generated, then $M = 0.$ A maximal ideal is a prime ideal and so one has :$\operatorname\left(R\right) = \bigcap_ \mathfrak \subset \operatorname\left(R\right)$ where the intersection on the left is called the nilradical of ''R''. As it turns out, $\operatorname\left(R\right)$ is also the set of nilpotent elements of ''R''. If ''R'' is an Artinian ring, then $\operatorname\left(R\right)$ is nilpotent and $\operatorname\left(R\right) = \operatorname\left(R\right)$. (Proof: first note the DCC implies $J^n = J^$ for some ''n''. If (DCC) $\mathfrak \supsetneq \operatorname\left(J^n\right)$ is an ideal properly minimal over the latter, then $J \cdot \left(\mathfrak/\operatorname\left(J^n\right)\right) = 0$. That is, $J^n \mathfrak = J^ \mathfrak = 0$, a contradiction.)

# Extension and contraction of an ideal

Let ''A'' and ''B'' be two
commutative ring In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
s, and let ''f'' : ''A'' → ''B'' be a ring homomorphism. If $\mathfrak$ is an ideal in ''A'', then $f\left(\mathfrak\right)$ need not be an ideal in ''B'' (e.g. take ''f'' to be the inclusion of the ring of integers Z into the field of rationals Q). The extension $\mathfrak^e$ of $\mathfrak$ in ''B'' is defined to be the ideal in ''B'' generated by $f\left(\mathfrak\right)$. Explicitly, :$\mathfrak^e = \Big\$ If $\mathfrak$ is an ideal of ''B'', then $f^\left(\mathfrak\right)$ is always an ideal of ''A'', called the contraction $\mathfrak^c$ of $\mathfrak$ to ''A''. Assuming ''f'' : ''A'' → ''B'' is a ring homomorphism, $\mathfrak$ is an ideal in ''A'', $\mathfrak$ is an ideal in ''B'', then: * $\mathfrak$ is prime in ''B'' $\Rightarrow$ $\mathfrak^c$ is prime in ''A''. * $\mathfrak^ \supseteq \mathfrak$ * $\mathfrak^ \subseteq \mathfrak$ It is false, in general, that $\mathfrak$ being prime (or maximal) in ''A'' implies that $\mathfrak^e$ is prime (or maximal) in ''B''. Many classic examples of this stem from algebraic number theory. For example, embedding $\mathbb \to \mathbb\left\lbrack i \right\rbrack$. In $B = \mathbb\left\lbrack i \right\rbrack$, the element 2 factors as $2 = \left(1 + i\right)\left(1 - i\right)$ where (one can show) neither of $1 + i, 1 - i$ are units in ''B''. So $\left(2\right)^e$ is not prime in ''B'' (and therefore not maximal, as well). Indeed, $\left(1 \pm i\right)^2 = \pm 2i$ shows that $\left(1 + i\right) = \left(\left(1 - i\right) - \left(1 - i\right)^2\right)$, $\left(1 - i\right) = \left(\left(1 + i\right) - \left(1 + i\right)^2\right)$, and therefore $\left(2\right)^e = \left(1 + i\right)^2$. On the other hand, if ''f'' is
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...
and $\mathfrak \supseteq \ker f$ then: * $\mathfrak^=\mathfrak$ and $\mathfrak^=\mathfrak$. * $\mathfrak$ is a
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together wi ...
in ''A'' $\Leftrightarrow$ $\mathfrak^e$ is a prime ideal in ''B''. * $\mathfrak$ is a maximal ideal in ''A'' $\Leftrightarrow$ $\mathfrak^e$ is a maximal ideal in ''B''. Remark: Let ''K'' be a
field extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ' ...
of ''L'', and let ''B'' and ''A'' be the rings of integers of ''K'' and ''L'', respectively. Then ''B'' is an integral extension of ''A'', and we let ''f'' be the
inclusion map In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B: \iota : A\rightarrow B, \qquad \iota ...
from ''A'' to ''B''. The behaviour of a
prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together wi ...
$\mathfrak = \mathfrak$ of ''A'' under extension is one of the central problems of
algebraic number theory Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ...
. The following is sometimes useful: a prime ideal $\mathfrak$ is a contraction of a prime ideal if and only if $\mathfrak = \mathfrak^$. (Proof: Assuming the latter, note $\mathfrak^e B_ = B_ \Rightarrow \mathfrak^e$ intersects $A - \mathfrak$, a contradiction. Now, the prime ideals of $B_$ correspond to those in ''B'' that are disjoint from $A - \mathfrak$. Hence, there is a prime ideal $\mathfrak$ of ''B'', disjoint from $A - \mathfrak$, such that $\mathfrak B_$ is a maximal ideal containing $\mathfrak^e B_$. One then checks that $\mathfrak$ lies over $\mathfrak$. The converse is obvious.)

# Generalizations

Ideals can be generalized to any monoid object $\left(R,\otimes\right)$, where $R$ is the object where the
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ar ...
structure has been forgotten. A left ideal of $R$ is a subobject $I$ that "absorbs multiplication from the left by elements of $R$"; that is, $I$ is a left ideal if it satisfies the following two conditions: # $I$ is a subobject of $R$ # For every $r \in \left(R,\otimes\right)$ and every $x \in \left(I, \otimes\right)$, the product $r \otimes x$ is in $\left(I, \otimes\right)$. A right ideal is defined with the condition "$r \otimes x \in \left(I, \otimes\right)$" replaced by "'$x \otimes r \in \left(I, \otimes\right)$". A two-sided ideal is a left ideal that is also a right ideal, and is sometimes simply called an ideal. When $R$ is a commutative monoid object respectively, the definitions of left, right, and two-sided ideal coincide, and the term ideal is used alone. An ideal can also be thought of as a specific type of -module. If we consider $R$ as a left $R$-module (by left multiplication), then a left ideal $I$ is really just a left sub-module of $R$. In other words, $I$ is a left (right) ideal of $R$ if and only if it is a left (right) $R$-module which is a subset of $R$. $I$ is a two-sided ideal if it is a sub-$R$-bimodule of $R$. Example: If we let $R=\mathbb$, an ideal of $\mathbb$ is an abelian group which is a subset of $\mathbb$, i.e. $m\mathbb$ for some $m\in\mathbb$. So these give all the ideals of $\mathbb$.

*
Modular arithmetic In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his boo ...
* Noether isomorphism theorem *
Boolean prime ideal theorem In mathematics, the Boolean prime ideal theorem states that ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement for filters on sets is known as the ultrafilter lemma. Other theorems are obtained by consid ...
* Ideal theory *
Ideal (order theory) In mathematical order theory, an ideal is a special subset of a partially ordered set (poset). Although this term historically was derived from the notion of a ring ideal of abstract algebra, it has subsequently been generalized to a different noti ...
* Ideal norm * Splitting of prime ideals in Galois extensions * Ideal sheaf

# References

* Atiyah, M. F. and Macdonald, I. G., '' Introduction to Commutative Algebra'', Perseus Books, 1969, * * Michiel Hazewinkel, Nadiya Gubareni, Nadezhda Mikhaĭlovna Gubareni, Vladimir V. Kirichenko. ''Algebras, rings and modules''. Volume 1. 2004. Springer, 2004. *