principal ideal ring

In this article the focus is on the more general concept of a principal ideal ring which is not necessarily a domain.

For this reason, uniserial was used to mean "Artinian principal ideal ring" even as recently as the 1970s.

The Zariski-Samuel theorem determines the structure of a commutative principal ideal rings.

The distinction is that a principal ideal ring may have zero divisors whereas a principal ideal domain cannot.

A ring in which every ideal is principal is called principal, or a principal ideal ring.

T. Hungerford, On the structure of principal ideal rings, Pacific J. Math.

Then C is an ideal in R, and hence principal, since R is a principal ideal ring.

It has been generalized to other structures like polynomials over principal ideal rings or polynomial rings, and also some classes of non-commutative rings and algebras.

Köthe showed that the modules of Artinian principal ideal rings (which are a special case of serial rings) are direct sums of cyclic submodules.

Every valuation ring is a uniserial ring, and all Artinian principal ideal rings are serial rings, as is illustrated by semisimple rings.

A commutative principal ideal ring which is also an integral domain is said to be a principal ideal domain (PID).

If R is a right principal ideal ring, then it is certainly a right Noetherian ring, since every right ideal is finitely generated.

More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors (e.g., Bourbaki) refer to PIDs as principal rings.

Later, Cohen and Kaplansky determined that a commutative ring R has this property for its modules if and only if R is an Artinian principal ideal ring.

The concept and algorithms of Gröbner bases have also been generalized to ideals over various ring, commutative or not, like polynomial rings over a principal ideal ring or Weyl algebras.

When this is satisfied for both left and right ideals, such as the case when R is a commutative ring, R can be called a principal ideal ring, or simply principal ring.