in this paper $s$ is a monoid with a left zero and $a_s$ (or $a$) is a unitary right $s$-act. it is shown that a monoid $s$ is right perfect (semiperfect) if and only if every (finitely generated) strongly flat right $s$-act is quasi-projective. also it is shown that if every right $s$-act has a unique zero element, then the existence of a quasi-projective cover for each right act implies that ...

Proof. Let L/F be an algebraic extension. Let f : L −→ L be a homomorphism fixing F . Recall that field homomorphisms are always injective, it remains to show that it is surjective. Let a ∈ L. As L/F is algebraic, there exists a1, . . . , ad ∈ F such that a satisfy p(x) = x + a1xd−1 + . . .+ ad. Let S = {s ∈ L : p(s) = 0}. As f is a homomorphism fixing the coefficients of the polynomial p(x), i...

Alexander duality is made into a functor which extends the notion for monomial ideals to any finitely generated N-graded module. The functors associated with Alexander duality provide a duality on the level of free and injective resolutions, and numerous Bass and Betti number relations result as corollaries. A minimal injective resolution of a module M is equivalent to the injective resolution ...

B efore meiotic divisions, chromosomes form crossovers, which represent the reciprocal exchanges between the DNA molecules of the homolo-gous partner chromosomes. These crossovers aren't just important for reassorting genetic traits; they're essential for the correct alignment and segregation of meiotic chromosomes. To ensure that every homologous chromosome pair forms at least one cross-over—a...

let r be a commutative noetherian ring. we study the behavior of injectiveand at dimension of r-modules under the functors homr(-,-) and -×r-.

In this paper $S$ is a monoid with a left zero and $A_S$ (or $A$) is a unitary right $S$-act. It is shown that a monoid $S$ is right perfect (semiperfect) if and only if every (finitely generated) strongly flat right $S$-act is quasi-projective. Also it is shown that if every right $S$-act has a unique zero element, then the existence of a quasi-projective cover for each right act implies that ...

Decomposable mappings from the space of symmetric k-fold tensors over E, ⊗ s,k E, to the space of k-fold tensors over F , ⊗ s,k F , are those linear operators which map nonzero decomposable elements to nonzero decomposable elements. We prove that any decomposable mapping is induced by an injective linear operator between the spaces on which the tensors are defined. Moreover, if the decomposable...

We study the injective envelope I(X) of an operator space X, showing amongst other things that it is a self-dual C∗module. We describe the diagonal corners of the injective envelope of the canonical operator system associated with X. We prove that if X is an operator A-B-bimodule, then A and B can be represented completely contractively as subalgebras of these corners. Thus, the operator algebr...

A Banach space is injective (resp. a (Pi space) if every isomorphic (resp. isometric) imbedding of it in an arbitrary Banach space Y is the range of a bounded (resp. norm-one) linear projection defined on Y. In §1 we study linear topological properties of injective Banach spaces and the spaces C(S) themselves; in §2 we study their conjugate spaces. (Throughout, " S " denotes an arbitrary compac...