ON FINITE PRINCIPAL IDEAL RINGS
Jilyana Cazaran, Andrei V. Kelarev · eCite Digital Repository (University of Tasmania) · 1999
We find new conditions sucient for a tensor product R S and a quotient ring Q/I to be a finite commutative principal ideal ring, where Q is a polynomial ring and I is an ideal of Q generated by univariate polynomials. 1. Main Results Finite commutative rings are interesting objects of ring theory and have many applications in combinatorics. For these applications it is often important to know when a ring is a principal ideal ring. Let us give only one example. Many classical error-correcting codes are ideals in finite commutative rings. The existence of single generators in ideals is important for computer storage as well as for encoding and deco ding algorithms (see (9)). If we want to use certain ring constructions in combinatorial applications of finite rings, then a natural question arises of when a ring construction is a principal ideal ring. This question has been considered in the literature for several ring constructions. For example, a complete description of commutativ e semigroup rings which are PIR's was obtained in (5). All graded commutative principal ideal rings were described in (4). This paper is devoted to two ring constructions which are important, general and lead to interesting results. All rings considered are commutative and have identity elements. We write for Z. For any ring R and prime p, the p-component of R is defined by R p ={r2 R| p k r = 0 for some positive integer k}. Let R be an arbitrary ring, p a prime, and let f2 R(x ). Denote by f the image of f in R(x)/pR(x). We say that f is squarefree (irreducible) modulo p if f is squarefree (respectively, irreducible). A Galois ring GR(p m ,r) is a ring of the form (Z/p m Z)(x)/(f(x)), where p is a prime, m an integer, and f(x)2Z/p m Z(x) is a monic polynomial of degree r which is irreducible modulo p.