In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a principal right (left) ideal ring is a ring ''R'' in which every right (left) ideal is of the form ''xR'' (''Rx'') for some element ''x'' of ''R''. (The right and left ideals of this form, generated by one element, are called
principal ideal
In mathematics, specifically ring theory, a principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R. The term also has another, similar meaning in order theory, where it ...
s.) When this is satisfied for both left and right ideals, such as the case when ''R'' is a
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 sp ...
, ''R'' can be called a principal ideal ring, or simply principal ring.
If only the
finitely generated right ideals of ''R'' are principal, then ''R'' is called a right Bézout ring. Left Bézout rings are defined similarly. These conditions are studied in domains as
Bézout domain In mathematics, a Bézout domain is a form of a Prüfer domain. It is an integral domain in which the sum of two principal ideals is again a principal ideal. This means that for every pair of elements a Bézout identity holds, and that every fini ...
s.
A commutative principal ideal ring which is also an
integral domain
In mathematics, specifically abstract algebra, 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 set ...
is said to be 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 principal, ...
'' (PID). In this article the focus is on the more general concept of a principal ideal ring which is not necessarily a domain.
General properties
If ''R'' is a principal right ideal ring, then it is certainly a right
Noetherian ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noether ...
, since every right ideal is finitely generated. It is also a right Bézout ring since all finitely generated right ideals are principal. Indeed, it is clear that principal right ideal rings are exactly the rings which are both right Bézout and right Noetherian.
Principal right ideal rings are closed under finite
direct product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
s. If
, then each right ideal of ''R'' is of the form
, where each
is a right ideal of ''R''
i. If all the ''R''
i are principal right ideal rings, then ''A''
i=''x''
i''R''
i, and then it can be seen that
. Without much more effort, it can be shown that right Bézout rings are also closed under finite direct products.
Principal right ideal rings and right Bézout rings are also closed under quotients, that is, if ''I'' is a proper ideal of principal right ideal ring ''R'', then the quotient ring ''R/I'' is also principal right ideal ring. This follows readily from the
isomorphism theorem
In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist f ...
s for rings.
All properties above have left analogues as well.
Commutative examples
1. The
ring of integers
In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often deno ...
:
2. The
integers modulo ''n'':
.
3. Let
be rings and
. Then ''R'' is a principal ring if and only if ''R''
''i'' is a principal ring for all ''i''.
4. The localization of a principal ring at any
multiplicative subset In abstract algebra, a multiplicatively closed set (or multiplicative set) is a subset ''S'' of a ring ''R'' such that the following two conditions hold:
* 1 \in S,
* xy \in S for all x, y \in S.
In other words, ''S'' is closed under taking finite ...
is again a principal ring. Similarly, any quotient of a principal ring is again a principal ring.
5. Let ''R'' be a
Dedekind domain
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily ...
and ''I'' be a nonzero ideal of ''R''. Then the quotient ''R''/''I'' is a principal ring. Indeed, we may factor ''I'' as a product of prime
powers:
, and by 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 ...
, so it suffices to see that each
is a principal ring. But
is isomorphic to the quotient
of the
discrete valuation ring
In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal.
This means a DVR is an integral domain ''R'' which satisfies any one of the following equivalent conditions:
# ''R'' ...
and, being a quotient of a principal ring, is itself a principal ring.
6. Let ''k'' be a finite field and put