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.

Let K and X be compact plane sets such that K X. Let P(K)be the uniform closure of polynomials on K. Let R(K) be the closure of rationalfunctions K with poles o K. Dene P(X;K) and R(X;K) to be the uniformalgebras of functions in C(X) whose restriction to K belongs to P(K) and R(K),respectively. Let CZ(X;K) be the Banach algebra of functions f in C(X) suchthat fjK = 0. In this paper, we show th...

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.

0.1. Endomorphisms of elliptic curves. Recall that a homomorphism of complex elliptic curves is just a holomorphic map E1 → E2 which preserves the origin. (It turns out that this condition is enough to force it to be a homomorphism of groups in the usual sense; why?) An isogeny of elliptic curves is a homomorphism whose kernel is a finite subgroup of E1. In fact the kernel of a homomorphism of ...

Let B be a strictly real commutative real Banach algebra with the carrier space Φ B. If A is a commutative real Banach algebra, then we give a representation of a ring homomorphism ρ : A → B, which needs not be linear nor continuous. If A is a commutative complex Banach algebra, then ρ(A) is contained in the radical of B. 1. Introduction and results. Ring homomorphisms are mappings between two ...

Let $R$ be a ring and $M$ a right $R$-module. We call $M$, coretractable relative to $overline{Z}(M)$ (for short, $overline{Z}(M)$-coretractable) provided that, for every proper submodule $N$ of $M$ containing $overline{Z}(M)$, there is a nonzero homomorphism $f:dfrac{M}{N}rightarrow M$. We investigate some conditions under which the two concepts coretractable and $overline{Z}(M)$-coretractable...

LetA be a finite-dimensional commutative algebra over the complex numbers with nonzero multiplicative identity element e. Let M denote the set of homomorphisms from A onto the complex numbers. Thus we get a homomorphism from A into the algebra of complex-valued functions on M, with respect to pointwise addition and multiplication of functions, by sending an element x of A to the function on M w...

let $r$ be a 2-torsion free ring and $u$ be a square closed lie ideal of $r$. suppose that $alpha, beta$ are automorphisms of $r$. an additive mapping $delta: r longrightarrow r$ is said to be a jordan left $(alpha,beta)$-derivation of $r$ if $delta(x^2)=alpha(x)delta(x)+beta(x)delta(x)$ holds for all $xin r$. in this paper it is established that if $r$ admits an additive mapping $g : rlongrigh...

Let be a commutative ring with 1 and left unitary . In this papers we introduced studied concept P-small compressible (An is said to if can embedded in every of it nonzero submodule Equivalently, there exists monomorphism , retractable for non-zero homomorphism whenever as generalization respectively give some their advantages characterizations examples.

A topological algebra is a complex associative algebra which is also a Hausdorff topological vector space such that the multiplication is separately continuous. A locally convex algebra is a topological algebra whose topology is determined by a family of seminorms. A complete metrizable locally convex algebra is called a B0-algebra. The topology of a B0-algebra A may be given by a countable fam...