It is shown that a commutative Bézout ring R with compact minimal prime spectrum is an elementary divisor ring if and only if so is R/L for each minimal prime ideal L. This result is obtained by using the quotient space pSpec R of the prime spectrum of the ring R modulo the equivalence generated by the inclusion. When every prime ideal contains only one minimal prime, for instance if R is arith...

Keigher showed that quasi-prime ideals in differential commutative rings are analogues of prime ideals in commutative rings. In that direction, he introduced and studied new types of differential rings using quasi-prime ideals of a differential ring. In the same sprit, we define and study two new types of differential rings which lead to the mirrors of the corresponding results on von Neumann r...

We investigate the structure of prime ideals of finite height in polynomial extension rings of a commutative unitary ring R. We consider the question of finite generation of such prime ideals. The valuative dimension of prime ideals of R plays an important role in our considerations. If X is an infinite set of indeterminates over R, we prove that every prime ideal of R[X] of finite height is fi...

We describe the structure of monoid natural-valued monotone functions on an arbitrary poset. For this we provide a presentation, characterization prime elements, and description its convex hull. also study associated ring, proving that it is normal, thus Cohen-Macaulay. determine Cohen-Macaulay type, characterize Gorenstein property, Gröbner basis defining ideal. Then apply these results to qua...

The set of all endomorphisms over -module is a non-empty denoted by . From we can construct the ring addition and composition function. prime ideal an which satisfies properties like numbers. In this paper, take integer number module such that ring. Furthermore, show existences on We also applied property to

In this paper, it has been proved that for a Noetherian ring R and an automorphism σ of R, an associated prime ideal of R[x, σ] or R[x, x−1, σ] is the extension of its contraction to R and this contraction is the intersection of the orbit under σ of some associated prime ideal of R. The same statement is true for minimal prime ideals also. It has also been proved that for a Noetherian Q-algebra...