ideal (ring theory)
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, and more specifically in ring theory, an ideal of a ring is a special
subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), 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 a ...
of its elements. Ideals generalize certain subsets of the
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
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 group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
, 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 ...
can be used to construct a quotient group. Among the integers, the ideals correspond one-for-one with the non-negative integers: 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 ideals 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 (mathematics), 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 ...
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 thes ...
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 is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
). 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 In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral do ...
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 and especially
Emmy Noether Amalie Emmy Noether (23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She also proved Noether's theorem, Noether's first and Noether's second theorem, second theorems, which ...
.


Definitions

Given a ring , a left ideal is a subset of that is a
subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation  ...
of the additive group of R that "absorbs multiplication from the left by elements of "; that is, I is a left ideal if it satisfies the following two conditions: # (I,+) is a
subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation  ...
of , # For every r \in R and every , the product r x is in . In other words, a left ideal is a left submodule of , considered as a left module over itself. A right ideal is defined similarly, with the condition rx\in I replaced by . A two-sided ideal is a left ideal that is also a right ideal. If the ring is
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...
, the three definitions are the same, and one talks simply of an ideal. In the non-commutative case, "ideal" is often used instead of "two-sided ideal". If is a left, right or two-sided ideal, the relation x \sim y if and only if :x-y\in I is an equivalence relation on , and the set of equivalence classes forms a left, right or bi module denoted R/I and called the '' quotient'' of by . (It is an instance of a congruence relation and is a generalization of
modular arithmetic In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to mo ...
.) If the ideal is two-sided, R/I is a ring, and the function :R\to R/I that associates to each element of its equivalence class is a surjective ring homomorphism that has the ideal as its kernel. Conversely, the kernel of a ring homomorphism is a two-sided ideal. Therefore, ''the two-sided ideals are exactly the kernels of ring homomorphisms.''


Note on convention

By convention, a ring has the multiplicative identity. But some authors do not require a ring to have the multiplicative identity; i.e., for them, a ring is a rng. For a rng , a left ideal is a with the additional property that rx is in for every r \in R and every x \in I. (Right and two-sided ideals are defined similarly.) For a ring, an ideal (say a left ideal) is rarely a subring; since a subring shares the same multiplicative identity with the ambient ring , if were a subring, for every r \in R, we have r = r 1 \in I; i.e., I = R. The notion of an ideal does not involve associativity; thus, an ideal is also defined for
non-associative ring A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative. That is, an algebraic structure ''A'' is a non-associative algebra over a field ''K'' if ...
s (often without the multiplicative identity) such as a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
.


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 (1) since it is precisely the two-sided ideal generated (see below) by the unity . Also, the set \ consisting of only the additive identity 0''R'' forms a two-sided ideal called the zero ideal and is denoted by .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). 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 = (r u^) u \in \mathfrak for every . Typically there are plenty of proper ideals. In fact, if ''R'' is a skew-field, then (0), (1) are its only ideals and conversely: that is, a nonzero ring ''R'' is a skew-field if (0), (1) 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 = (1); i.e., yx = 1 for some nonzero . Likewise, zy = 1 for some nonzero z. Then z = z(yx) = (zy)x = x.) * The even
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s form an ideal in the ring \mathbb of all integers, since the sum of any two even integers is even, and the product of any integer with an even integer is also even; this ideal is usually denoted by . More generally, the set of all integers divisible by a fixed integer n is an ideal denoted . In fact, every non-zero ideal of the ring \mathbb is generated by its smallest positive element, as a consequence of Euclidean division, so \mathbb is a principal ideal domain. * The set of all polynomials with real coefficients that are divisible by the polynomial x^2+1 is an ideal in the ring of all real-coefficient polynomials . * Take a ring R and positive integer . For each , the set of all n\times n matrices with entries in R whose i-th row is zero is a right ideal in the ring M_n(R) of all n\times n matrices with entries in . It is not a left ideal. Similarly, for each , the set of all n\times n matrices whose j-th ''column'' is zero is a left ideal but not a right ideal. * The ring C(\mathbb) of all continuous functions f from \mathbb to \mathbb under pointwise multiplication contains the ideal of all continuous functions f such that . Another ideal in C(\mathbb) is given by those functions that 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 . * A ring is called a simple ring if it is nonzero and has no two-sided ideals other than . 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(f) = f^(0_S) is a two-sided ideal of . By definition, , and thus if S is not the zero ring (so ), then \ker(f) is a proper ideal. More generally, for each left ideal ''I'' of ''S'', the pre-image f^(I) is a left ideal. If ''I'' is a left ideal of ''R'', then f(I) is a left ideal of the subring f(R) of ''S'': unless ''f'' is surjective, f(I) need not be an ideal of ''S''; see also . * Ideal correspondence: Given a surjective ring homomorphism , 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(I) and the pre-image . 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). * If ''M'' is a left ''R''- module and S \subset M a subset, then the annihilator \operatorname_R(S) = \ of ''S'' is a left ideal. Given ideals \mathfrak, \mathfrak of a commutative ring ''R'', the ''R''-annihilator of (\mathfrak + \mathfrak)/\mathfrak is an ideal of ''R'' called the ideal quotient of \mathfrak by \mathfrak and is denoted by ; it is an instance of
idealizer In abstract algebra, the idealizer of a subsemigroup ''T'' of a semigroup ''S'' is the largest subsemigroup of ''S'' in which ''T'' is an Semigroup#Subsemigroups and ideals, ideal. Such an idealizer is given by :\mathbb_S(T)=\. In ring theory, if ...
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 . 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 Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least on ...
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 = (0) and , 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 . 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. ). The principal two-sided ideal RxR is often also denoted by . If X = \ is a finite set, then RXR is also written as . * There is a bijective correspondence between ideals and congruence relations (equivalence relations that respect the ring structure) on the ring: Given an ideal I of a ring , let x\sim y if . Then \sim is a congruence relation on . Conversely, given a congruence relation \sim on , let . Then I is an ideal of .


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 is called a maximal ideal if there exists no other proper ideal with a proper subset of . 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. * Zero ideal: the ideal \. * Unit ideal: the whole ring (being the ideal generated by 1). * Prime ideal: A proper ideal I is called a prime ideal if for any a and b in , if ab is in , then at least one of a and b is in . The factor ring of a prime ideal is a prime ring in general and is an
integral domain In mathematics, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibilit ...
for commutative rings. * Radical ideal or semiprime ideal: A proper ideal is called radical or semiprime if for any in R, if is in for some , then is in . 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. ...
: An ideal is called a primary ideal if for all and in , if is in , then at least one of and is in for some
natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
. 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 Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by John ...
left module. * Irreducible ideal: An ideal is said to be irreducible if it cannot be written as an intersection of ideals that properly contain it. * Comaximal ideals: Two ideals , are said to be comaximal if x + y = 1 for some x \in I and . * 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. * Perfect ideal: A proper ideal in a Noetherian ring R is called a perfect ideal if its grade equals the projective dimension of the associated quotient ring, . A perfect ideal is unmixed. * Unmixed ideal: A proper ideal in a Noetherian ring R is called an unmixed ideal (in height) if the height of is equal to the height of every associated prime of R/I. (This is stronger than saying that R/I is equidimensional. See also equidimensional ring. Two other important terms using "ideal" are not always ideals of their ring. See their respective articles for details: *
Fractional ideal In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral do ...
: This is usually defined when R is a commutative domain with quotient field K. Despite their names, fractional ideals are not necessarily ideals. A fractional ideal of R is an R-submodule of K for which there exists a non-zero r \in R such that rI \subseteq R. If the fractional ideal is contained entirely in R, then it is truly an ideal of R. * Invertible ideal: Usually an invertible ideal is defined as a fractional ideal for which there is another fractional ideal such that . Some authors may also apply "invertible ideal" to ordinary ring ideals and with in rings other than domains.


Ideal operations

The sum and product of ideals are defined as follows. For \mathfrak and , 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 . Note \mathfrak + \mathfrak is the smallest left (resp. right) ideal containing both \mathfrak and \mathfrak (or the union ), while the product \mathfrak\mathfrak is contained in the intersection of \mathfrak and . The distributive law holds for two-sided ideals , * , * . If a product is replaced by an intersection, a partial distributive law holds: : \mathfrak \cap (\mathfrak + \mathfrak) \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 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 = (1) * \mathfrak is generated by elements that form a regular sequence modulo . (More generally, the difference between a product and an intersection of ideals is measured by the Tor functor: .) An integral domain is called a Dedekind domain if for each pair of ideals \mathfrak \subset \mathfrak, there is an ideal \mathfrak such that . 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.


Examples of ideal operations

In \mathbb we have : (n)\cap(m) = \operatorname(n,m)\mathbb since (n)\cap(m) is the set of integers that are divisible by both n and . Let R = \mathbb ,y,z,w/math> and let . Then, * \mathfrak + \mathfrak = (z,w, x+z, y+w) = (x, y, z, w) and \mathfrak + \mathfrak = (z, w, x) * \mathfrak\mathfrak = (z(x + z), z(y + w), w(x + z), w(y + w))= (z^2 + xz, zy + wz, wx + wz, wy + w^2) * \mathfrak\mathfrak = (xz + z^2, zw, xw + zw, w^2) * \mathfrak \cap \mathfrak = \mathfrak\mathfrak while \mathfrak \cap \mathfrak = (w, xz + z^2) \neq \mathfrak\mathfrak In the first computation, we see the general pattern for taking the sum of two finitely generated ideals, it is the ideal generated by the union of their generators. In the last three we observe that products and intersections agree whenever the two ideals intersect in the zero ideal. These computations can be checked using Macaulay2.


Radical of a ring

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(R) 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(M) = R/\operatorname(x) \simeq M, meaning \operatorname(M) is a maximal ideal. Conversely, if \mathfrak is a maximal ideal, then \mathfrak is the annihilator of the simple ''R''-module . 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 , then ''M'' does not admit a
maximal submodule In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals ...
, since if there is a maximal submodule , J \cdot (M/L) = 0 and so , 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 . A maximal ideal is a prime ideal and so one has : \operatorname(R) = \bigcap_ \mathfrak \subset \operatorname(R) where the intersection on the left is called the nilradical of ''R''. As it turns out, \operatorname(R) is also the set of nilpotent elements of ''R''. If ''R'' is an Artinian ring, then \operatorname(R) is nilpotent and . (Proof: first note the DCC implies J^n = J^ for some ''n''. If (DCC) \mathfrak \supsetneq \operatorname(J^n) is an ideal properly minimal over the latter, then J \cdot (\mathfrak/\operatorname(J^n)) = 0. That is, , a contradiction.)


Extension and contraction of an ideal

Let ''A'' and ''B'' be two
commutative ring In mathematics, a commutative ring is a Ring (mathematics), 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 prope ...
s, and let ''f'' : ''A'' → ''B'' be a ring homomorphism. If \mathfrak is an ideal in ''A'', then f(\mathfrak) 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 . Explicitly, : \mathfrak^e = \Big\ If \mathfrak is an ideal of ''B'', then f^(\mathfrak) 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 In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group (mathematics), group that is a subgroup. When some object X is said to be embedded in another object Y ...
. In B = \mathbb\left\lbrack i \right\rbrack, the element 2 factors as 2 = (1 + i)(1 - i) where (one can show) neither of 1 + i, 1 - i are units in ''B''. So (2)^e is not prime in ''B'' (and therefore not maximal, as well). Indeed, (1 \pm i)^2 = \pm 2i shows that , , and therefore . On the other hand, if ''f'' is surjective and \mathfrak \supseteq \ker f then: * \mathfrak^=\mathfrak and . * \mathfrak is a prime ideal 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 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 from ''A'' to ''B''. The behaviour of a prime ideal \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 ob ...
. The following is sometimes useful: a prime ideal \mathfrak is a contraction of a prime ideal if and only if . (Proof: Assuming the latter, note \mathfrak^e B_ = B_ \Rightarrow \mathfrak^e intersects , a contradiction. Now, the prime ideals of B_ correspond to those in ''B'' that are disjoint from . Hence, there is a prime ideal \mathfrak of ''B'', disjoint from , such that \mathfrak B_ is a maximal ideal containing . One then checks that \mathfrak lies over . The converse is obvious.)


Generalizations

Ideals can be generalized to any monoid object , where R is the object where the monoid structure has been forgotten. A left ideal of R is a subobject I that "absorbs multiplication from the left by elements of "; that is, I is a left ideal if it satisfies the following two conditions: # I is a subobject of R # For every r \in (R,\otimes) and every , the product r \otimes x is in . 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. 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.


See also

*
Modular arithmetic In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to mo ...
* Noether isomorphism theorem * Boolean prime ideal theorem * Ideal theory *
Ideal (order theory) In mathematics, 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 ...
* Ideal norm * Splitting of prime ideals in Galois extensions * Ideal sheaf


Notes


References

* * * * * *


External links

* {{Authority control Algebraic structures Commutative algebra Algebraic number theory