Finite Integral Domain Research Articles

Zero Divisor Graphs of Upper Triangular Matrix Rings

Let R be a commutative ring with identity 1 ≠ 0 and T be the ring of all n × n upper triangular matrices over R. In this paper, we describe the zero divisor graph of T. Some basic graph theory properties of are given, including determination of the girth and diameter. The structure of is discussed, and bounds for the number of edges are given. In the case that R is a finite integral domain and n = 2, the structure of is fully described and an explicit formula for the number of edges is given.

On the FIP Property for Extensions of Commutative Rings #

ABSTRACT A (unital) extension R ⊆ T of (commutative) rings is said to have FIP (respectively be a minimal extension) if there are only finitely many (respectively no) rings S such that R ⊂ S ⊂ T. Transfer results for the FIP property for extensions of Nagata rings are obtained, including the following fact: if R ⊂ T is a (module-) finite minimal ring extension, then R(X)⊂T(X) also is a (module-) finite minimal ring extension. The assertion obtained by replacing “is a (module-) finite minimal ring extension” with “has FIP” is valid if R is an infinite field but invalid if R is a finite field. A generalization of the Primitive Element Theorem is obtained by characterizing, for any field (more generally, any artinian reduced ring) R, the ring extensions R ⊆ T which have FIP; and, if R is any field K, by describing all possible structures of the (necessarily minimal) ring extensions appearing in any maximal chain of intermediate rings between K and any such T. Transfer of the FIP and “minimal extension” properties is given for certain pullbacks, with applications to constructions such as CPI-extensions. Various sufficient conditions are given for a ring extension of the form R ⊆ R[u], with u a nilpotent element, to have or not have FIP. One such result states that if R is a residually finite integral domain that is not a field and u is a nilpotent element belonging to some ring extension of R, then R ⊆ R[u] has FIP if and only if (0 : u) ≠ 0. The rings R having only finitely many unital subrings are studied, with complete characterizations being obtained in the following cases: char(R)>0; R an integral domain of characteristic 0; and R a (module-)finite extension of ℤ which is not an integral domain. In particular, a ring of the last-mentioned type has only finitely many unital subrings if and only if (ℤ:R)≠0. Some results are also given for the residually FIP property.

Construction of the Integral Closure of a Finite Integral Domain. II

Construction of the Integral Closure of a Finite Integral Domain. II

Construction of the integral closure of a finite integral domain. II

In a previous paper the problem of constructing the integral closure of a finite integral domain k [ x 1 , … , x n ] = k [ x ] k[{x_1}, \ldots ,{x_n}] = k[x] was considered. A reduction to the case d t k ( x ) / k = 1 , k ( x ) / k dtk(x)/k = 1,k(x)/k separable, and n = 2 n = 2 was made. A subsidiary problem was: if k [ x ] k[x] is not integrally closed, to find a y y in k ( x ) k(x) integral over k [ x ] k[x] but not in it. This was done for n = 2 n = 2 , but should have been done for arbitrary n n . The extra details are here given. For the convenience of the reader, the full argument is sketched.

Construction of the integral closure of a finite integral domain

The problem is to construct the integral closure of a finite integral domaink[x 1,..,x n ]. If the characteristicp is zero, this can be done for any explicitly given fieldk; ifp > 0, another condition must be imposed.

Differential Ideals in Rings of Finitely Generated Type

Let 6 be a commutative Noetherian ring containing the rational numbers, T a family of derivations mapping 6 into itself, and f a T-differential 6ideal (i. e., DE C f for every D C T). Theorem 1 shows that the associated prime ideals of f are also T-differential and that f has an irredundant primary decomposition into T-differential primary ideals. Theorem 2 shows that in studying a differential (i. e., T-differential for maximal T) prime ideal p in a finitely generated extension =k [xr, , xn] of a base field k one can localize, i. e., that p is differential in J if and only if p 6p is differential in the local ring 6p. Already here one needs more than the Noetherian condition and the study becomes essentially one for rings of finitely generated type, i. e., quotient rings of finite extensions of a base field k (of characteristic 0). We describe the further results for the case that j = k[x] is a finite integral domain or, equivalently, is the ring of an irreducible affine variety V over k. Theorem 3 shows that the simple subvarieties of V yield nondifferential ideals . By a lemma of Zariski, if p 6p is not differential, then the completion 6p of 6p is of the form O [Em]], where u C p p and is analytically independent over E. We say that V is analytically a product at or along an irreducible subvariety 't if the completion of O., where p =c ('t), is of the form just mentioned. Intuitively we think of V as somewhat like a cylinder built on the section u 0 of V. Then a component of the singular locus of V could not very well yield a non-differential ideal in 6; and the facts correspond to this intuition: Theorems 4 and 5 give the proof. Theorem 12, Corollary shows that V is analytically a product along a subvariety St if and only if the ideal of ' in the ring of V is not differential. While this is in some ways a satisfactory criterion (for a prime ideal to be differential), we are further concerned with the question whether the condition