Ideal RingEdit

An ideal ring is a ring in which every ideal is generated by a single element. In practice, most authors use the term principal ideal ring (PIR) to denote this property, and the phrase “ideal ring” or “two-sided principal ideal ring” is often employed when one wants to emphasize the two-sided nature of ideals in a noncommutative setting. The notion sits at the crossroads of ring theory and module theory: if every two-sided ideal of a ring R is principal, then every quotient R/I is also a principal-idealed object, and many questions about submodules of R or quotients by ideals reduce to questions about how a single generator behaves.

This topic is studied in both the commutative and noncommutative worlds, but the terminology and emphasis shift somewhat. In the commutative case, a ring in which every ideal is principal is typically just called a principal ideal ring; when the ring is also an integral domain, such rings are precisely principal ideal domains if one requires every ideal to be generated by one element. In the noncommutative setting, one must distinguish between left, right, and two-sided ideals, and the literature often reserves “principal ideal ring” for the two-sided version. The interplay with familiar objects such as fields, integer rings, and matrix rings helps illuminate what ideal rings look like in practice. To situate the topic, it is also useful to compare with related notions such as the Chinese remainder theorem, localization, and factorization phenomena in modules over such rings.

Definition and basic properties

Let R be a ring with unity. An ideal I ⊆ R is a subset closed under addition and under multiplication by any element of R from both sides. The ring R is called a two-sided principal ideal ring (or simply an ideal ring in the two-sided sense) if every two-sided ideal I is of the form I = Ra for some a ∈ R. When every left ideal is principal (I = Ra for some a) we speak of a left principal ring, and similarly for right principal rings. In the contexts used here, the focus is on the two-sided condition, which makes quotients and products behave coherently from the perspective of two-sided ideals.

Several immediate consequences follow: - Fields are the simplest examples, since the only ideals are {0} and the field itself, both principal. - If R is a two-sided PIR, then every quotient R/I is again a two-sided PIR for any ideal I; this follows because ideals in R/I correspond to ideals of R containing I, each of which is principal. - If R ≅ R1 × R2 is a finite direct product of two rings each with the PIR property, then R is a PIR, because the ideals split componentwise and are generated by suitable pairs of generators. - Simple rings (rings with only {0} and R as two-sided ideals) are PIR, since the only ideals are trivial and generated by 0 or 1.

In the commutative case, a PIR need not be a domain; rings like Z/nZ (for composite n) are PIRs that contain zero divisors. In the noncommutative world, matrix rings over division rings, such as M_n(D), are PIRs because their only two-sided ideals are {0} and the whole ring.

Examples

The ring of integers, Z, is a principal ideal ring: every ideal nZ is generated by a single integer n.
The ring of integers modulo n, Z, is a PIR for any positive integer n; its ideals correspond to divisors of n and are all principal.
A field F is a PIR, with only the trivial ideals {0} and F.
The polynomial ring in one indeterminate over a field, [[k[x]|k[x]]], is a PIR; every ideal is principal (generated by a single polynomial).
The ring of n-by-n matrices over a division ring, M_n(D), is a PIR because it is simple: its only two-sided ideals are {0} and the entire ring.
Rings like [[k[x, y]|k[x, y]]] with two independent indeterminates are typically not PIR, illustrating how the property breaks down when passing from a single variable to multivariable settings.

Quotients, products, and structure

PIRs enjoy a number of closure properties that make them convenient in constructive work: - Quotients: If R is a PIR, then any quotient R/I is a PIR for any ideal I, since ideals in R/I come from principal ideals in R. - Direct products: The finite direct product of PIRs is again a PIR. - Locality and decomposition: In the commutative setting, PIRs often arise as finite products of local components, reflecting how ideals decompose in a manner reminiscent of the Chinese remainder theorem.

In the commutative setting, principal ideal rings sit near the boundary between highly structured rings (like fields, PIDs, and Artinian principal rings) and more general rings. They intersect with classical factorization theory and, for integral domains, with the notion of unique factorization in a broader lattice of ideals. The study of modules over PIRs benefits from the fact that submodules of cyclic modules correspond to divisors of the generator, simplifying classification problems.

Noncommutative principal ideal rings

When ideals are considered in noncommutative rings, the landscape broadens. Simple rings, including matrix rings over division rings, provide a large and natural class of PIRs, since their two-sided ideals are limited to {0} and R. Other noncommutative PIRs arise by combining simple components with more intricate local structures, but care is needed: not every noncommutative ring with simple components is a PIR, and the lattice of ideals can be more complicated when left or right ideals are taken into account.

The interplay with modules remains central: for a PIR, cyclic modules R/I have a tractable structure, and many questions about submodules reduce to questions about principal ideals. This makes PIRs useful as a testing ground for ideas about generators, relations, and homomorphisms in the noncommutative setting, even as the broader theory of noncommutative rings explores far richer and more variable ideal lattices.

Controversies and debate

In practice, the usefulness of focusing on rings with every ideal generated by a single element is widely appreciated for its clarity and constructive nature. Proponents highlight several advantages: - Concrete calculability: Many questions about ideals, quotient rings, and maps reduce to explicit generators and relations, which makes algorithmic and computational work more feasible. - Pedagogical clarity: PIRs provide a clean stepping stone from fields and PIDs to more general ring theory, illustrating how ideals control structure. - Broad applicability: Classical examples such as Z, Z/nZ, and M_n(D) show that PIRs capture a useful diversity of algebraic behavior, from arithmetic to linear algebra.

Critics, however, point out that PIRs are, by construction, a relatively narrow slice of the universe of rings. The insistence that every ideal is generated by one element excludes many naturally occurring and important rings, including coordinate rings of most algebraic varieties in more than one variable, many algebras used in analysis and representation theory, and a large portion of noncommutative geometry. From this viewpoint, focusing on PIRs can miss phenomena such as: - Rich lattice of ideals and primary decompositions found in more general rings. - Homological and categorical properties that go beyond principal generators, such as projective resolutions, Ext and Tor groups, and derived category methods. - Rings that are not PIR but still exhibit tractable behavior locally or after localization.

From a practical angle, supporters argue that the PIR perspective often yields transparent proofs and explicit descriptions that can guide broader investigations, while critics caution against letting neat generators obscure deeper invariants and global structure.