We investigate Buchbaum and Eisenbud's construction of the second symmetric power $s_R(X)$ of a chain complex $X$ of modules over a commutative ring $R$. We state and prove a number of results from the folklore of the subject for which we know of no good direct references. We also provide several explicit computations and examples. We use this construction to prove the following vers...

a module $m$ is said to be coretractable if there exists a nonzero homomorphism of every nonzero factor of $m$ into $m$. we prove that all right (left) modules over a ring are coretractable if and only if the ring is morita equivalent to a finite product of local right and left perfect rings.

If (R,M) and (S,N) are quasilocal (commutative integral) domains and f : R → S is a (unital) ring homomorphism, then f is said to be a strong local homomorphism (resp., radical local homomorphism) if f (M) = N (resp., f (M) ⊆ N and for each x ∈ N , there exists a positive integer t such that xt ∈ f (M)). It is known that if f : R→ S is a strong local homomorphism where R is a pseudovaluation do...

the concepts of fuzzy semi-ideals of r with respect to h≤r and generalized fuzzy quotient rings are introduced. some properties of fuzzy semiideals are discussed. finally, several isomorphism theorems for generalized fuzzy quotient rings are established.

We discuss Matijevic–Roberts type theorem on strong F -regularity, F -purity, and Cohen–Macaulay F -injective (CMFI for short) property. Related to this problem, we also discuss the base change problem and the openness of loci of these properties. In particular, we define the notion of F -purity of homomorphisms using Radu–André homomorphisms, and prove basic properties of it. We also discuss a...

Recall that a continuous function $fcolon Xto Y$ between Tychonoff spaces is proper if and only if the Stone extension $f^{beta}colon beta Xtobeta Y$ takes remainder to remainder, in the sense that $f^{beta}[beta X-X]subseteq beta Y-Y$. We introduce the notion of ``taking remainder to remainder" to frames, and, using it, we define a frame homomorphism $hcolon Lto M$ to be $beta$-proper, $lambd...

Let $f: Arightarrow B$ be a ring homomorphism and let $J$ be an ideal of $B$. In this paper, we investigate the transfer of the property of coherence to the amalgamation $Abowtie^{f}J$. We provide necessary and sufficient conditions for $Abowtie^{f}J$ to be a coherent ring.

Let (M,ω) be a closed symplectic manifold and Ham(M,ω) the group of Hamiltonian diffeomorphisms of (M,ω). Then the Seidel homomorphism is a map from the fundamental group of Ham(M,ω) to the quantum homology ring QH∗(M ; Λ). Using this homomorphism we give a sufficient condition for when a nontrivial loop ψ in Ham(M,ω) determines a nontrivial loop ψ × idN in Ham(M ×N, ω ⊕ η), where (N, η) is a c...

This paper continues the investigation of the rst author begun in part one. The hereditary properties of n-homomorphism amenability for Banach algebras are investigated and the relations between n-homomorphism amenability of a Banach algebra and its ideals are found. Analogous to the character amenability, it is shown that the tensor product of two unital Banach algebras is n-homomorphism amena...

In this paper, we consider a class of connected oriented (with respect to Z/p) closed G-manifolds with a non-empty finite fixed point set, each of which is G-equivariantly formal, where G = Z/p and p is an odd prime. Using localization theorem and equivariant index, we give an explicit description of the mod p equivariant cohomology ring of such a G-manifold in terms of algebra. This makes ...