Let $R$ be a commutative ring with unity and $W=\{f(X)\in R[X]:f(0)=1\}$. We define $R\{X\}=W^{-1}R[X]$. We show that the maximal ideals of $R\{X\} $ are of the form $W^{-1}(M,X)$ where $M$ is a maximal ideal of $R$, and so if $R$ is finite dimensional, then $\dim R\{X\}=\dim R[X]$. We show that $R\{X\}$ is a Prüfer ring if and only if $R$ is a von Neumann regular ring, and so if $R\{X\}$ satisfies one of the Prüfer conditions, it satisfies all of them.

Let A be a commutative ring with identity, G be an abelian group, and consider the group ring AG. A ring A is called a generalized morphic ring (GM ring) if the annihilator of each element in A is principal. In this article, we showed that if AG is a GM ring, then so is A. The converse was proved to be false. We try to put some conditions on A or G to get the converse. Among many other results, we showed that if A is an Armendariz ring and G is a torsion free group, then AG is a GM ring if and only if A is. Moreover, if C-m denotes the multiplicative cyclic group of order m and Z(n) the ring of integers modulo n, we justified that the ring Z(n)C(m) is a GM ring if and only if, whenever p is a prime dividing gcd(n,m), then p(2) n. We also proved that for an integral domain D with char(D) =p, the group ring DCp is a GM ring.

In this article we relate the six Prufer conditions with the EM conditions. We use the EM-conditions to prove some cases of equivalence of the six Prufer conditions. We also use the Prufer conditions to answer some open problems concerning EM-rings.

A commutative ring with unity R is called an EM-ring if for any finitely generated ideal I there exist a in R and a finitely generated ideal J with Ann(J) = 0 and I = aJ. In this article it is proved that C(X) is an EM-ring if and only if for each U is an element of Coz (X), and each g is an element of C* (U) there is V is an element of Coz (X) such that U subset of V, (V) over bar = X, and g is continuously extendable on V. Such a space is called an EM-space. It is shown that EM-spaces include a large class of spaces as F-spaces and cozero complemented spaces. It is proved among other results that X is an EM-space if and only if the Stone-Cech compactification of X is.

Let [Formula: see text] be a commutative ring. A polynomial [Formula: see text] is an annihilating content (AC) polynomial if [Formula: see text] where [Formula: see text] and [Formula: see text] is a nonzerodivisor and [Formula: see text] is an EM-ring if each [Formula: see text] is an AC polynomial. In this paper, we investigate AC polynomials and EM-rings and extensions to modules. For example, we show that [Formula: see text] is an EM-ring if and only if every linear polynomial is an AC polynomial if and only if for each finitely generated ideal [Formula: see text] of [Formula: see text], [Formula: see text] where [Formula: see text] and [Formula: see text] is a finitely generated ideal of [Formula: see text] with [Formula: see text]. Special attention is given to the case when [Formula: see text] is Noetherian where such rings are characterized by the property that each associated prime is principal.

This is a survey for all the work done so far on EM-rings, their extensions, and some of their generalizations.

A commutative ring R with unityis called EM-Hermite if for each a, b is an element of R there exist c, d, f is an element of R such that a = cd,b = cf and the ideal (d, f) is regular in R. We showed in this article that R is a PP-ring if and only if the idealization R(+)R is an EM-Hermite ring if and only if R[x]/(x(n+1)) is an EM-Hermite ring for each n is an element of N. We generalize some results, and answer some questions in the literature.

A ring $R$ is called EM-Hermite if for each $a,b\in R$, there exist $% a_{1},b_{1},d\in R$ such that $a=a_{1}d,b=b_{1}d$ and the ideal $% (a_{1},b_{1})$ is regular. We give several characterizations of EM-Hermite rings analogue to those for K-Hermite rings, for example, $R$ is an EM-Hermite ring if and only if any matrix in $M_{n,m}(R)$ can be written as a product of a lower triangular matrix and a regular $m\times m$ matrix. We relate EM-Hermite rings to Armendariz rings, rings with a.c. condition, rings with property A, EM-rings, generalized morphic rings, and PP-rings. We show that for an EM-Hermite ring, the polynomial ring and localizations are also EM-Hermite rings, and show that any regular row can be extended to regular matrix. We relate EM-Hermite rings to weakly semi-Steinitz rings, and characterize the case at which every finitely generated $R$-module with finite free resolution of length 1 is free.

All rings R in this article are assumed to be commutative with unity 1 not equal 0. A ring R is called a GPF-ring if for every a is an element of R there exists a positive integer n such that the annihilator ideal Ann R (a(n)) is pure. We prove that for a ring R and an Abelian group G, if the group ring RG is a GPF ring then so is R. Moreover, if G is a finite Abelian group then vertical bar G vertical bar is a unit or a zero-divisor in R. We prove that if G is a group such that for every nontrivial subgroup H of G, [G : H] < infinity, then the group ring RG is a GPF-ring if and only if RH is a GPF-ring for each finitely generated subgroup H of G. It is proved that if R is a local ring and RG is a U group ring, then RG is a GPF-ring if and only if R is a GPF ring and p is an element of Nil (R). Finally, we prove that if R is a semisimple ring and G is a finite group such that vertical bar G vertical bar(-1) is an element of R, then RG is a GPF ring if and only if RG is a PF-ring.