Let $(R, m)$ be a commutative noetherian local ring and let $Gamma$ be a finite group. It is proved that if $R$ admits a dualizing module, then the group ring $Rga$ has a dualizing bimodule as well. Moreover, it is shown that a finitely generated $Rga$-module $M$ has generalized Gorenstein dimension zero if and only if it has generalized Gorenstein dimension zero as an $R$-module.

 A topoframe, denoted by $L_{ tau}$,  is a pair $(L, tau)$ consisting of a frame $L$ and a subframe $ tau $ all of whose elements are complementary elements in $L$. In this paper, we define and study the notions of a $tau $-real-continuous function on a frame $L$ and the set of real continuous functions $mathcal{R}L_tau $ as an $f$-ring. We show that $mathcal{R}L_{ tau}$ is actually a generali...

Let R be a commutative noetherian ring and Γ a finite group. In this paper,we study Gorenstein homological dimensions of modules with respect to a semi-dualizing module over the group ring  . It is shown that Gorenstein homological dimensions  of an  -RΓ module M with respect to a semi-dualizing module,  are equal over R and RΓ  .

Quantum yields for photochemical ring opening of six alkylcyclobutenes have been measured in hexane solution using 228-nm excitation, which selectively populates the lowest pi,R(3s) excited singlet states of these molecules and has been shown previously to lead to ring opening with clean conrotatory stereochemistry. The compounds studied in this work-1,2-dimethylcyclobutene (1), cis- and trans-...

A square matrix is said to be alternating-clean if it is the sum of an alternating matrix and an invertible matrix. In this paper, we determine all alternating-clean matrices over any division ring K. If K is not commutative, all matrices are alternating-clean, with the exception of the 1× 1 zero matrix. If K is commutative, all matrices are alternating-clean, with the exception of odd-size alt...

The purpose of this study was to identify relationships between core stability and various strength and power variables in strength and power athletes. National Collegiate Athletic Association Division I football players (height 184.0 +/- 7.1 cm, weight 100.5 +/- 22.4 kg) completed strength and performance testing before off-season conditioning. Subjects were tested on three strength variables ...

Let $f:A\rightarrow B$ be a ring homomorphism and $K$ an ideal of $B$. Many variations the notions clean nil-clean rings have been studied by variety authors. We investigate strongly $\pi$-regular clean-like properties amalgamation $A\bowtie^{f}K$ $A$ with $B$ along respect to $f$.

Let R be a commutative noetherian ring. We study the behavior of injectiveand at dimension of R-modules under the functors HomR(-,-) and -×R-.

We show that the lattice of all ideals of a ring $R$ can be embedded in the lattice of all its fuzzyideals in uncountably many ways. For this purpose, we introduce the concept of the generalizedcharacteristic function $chi _{s}^{r} (A)$ of a subset $A$ of a ring $R$ forfixed $r , sin [0,1] $ and show that $A$ is an ideal of $R$ if, and only if, its generalizedcharacteristic function $chi _{s}^{...

Let $R$ be a commutative ring. The purpose of this article is to introduce a new class of ideals of R called weakly irreducible ideals. This class could be a generalization of the families quasi-primary ideals and strongly irreducible ideals. The relationships between the notions primary, quasi-primary, weakly irreducible, strongly irreducible and irreducible ideals, in different rings, has bee...