Generalized Fitting modules and rings

Rings and modules

Proof. Consider the map g:A/I → C , a+I 7→ f (a). It is well defined: a+I = a′ +I implies a− a′ ∈ I implies f (a) = f (a′). The element a + I belongs to the kernel of g iff g(a + I) = f (a) = 0, i.e. a ∈ I , i.e. a + I = I is the zero element of A/I . Thus, ker(g) = 0. The image of g is g(A/I) = {f (a) : a ∈ A} = C . Thus, g is an isomorphism. The inverse morphism to g is given by f (a) 7→ a + I .

Graded Rings and Modules

1 Definitions Definition 1. A graded ring is a ring S together with a set of subgroups Sd, d ≥ 0 such that S = ⊕ d≥0 Sd as an abelian group, and st ∈ Sd+e for all s ∈ Sd, t ∈ Se. One can prove that 1 ∈ S0 and if S is a domain then any unit of S also belongs to S0. A homogenous ideal of S is an ideal a with the property that for any f ∈ a we also have fd ∈ a for all d ≥ 0. A morphism of graded r...

The generalized total graph of modules respect to proper submodules over commutative rings.

Let $M$ be a module over a commutative ring $R$ and let $N$ be a proper submodule of $M$. The total graph of $M$ over $R$ with respect to $N$, denoted by $T(Gamma_{N}(M))$, have been introduced and studied in [2]. In this paper, A generalization of the total graph $T(Gamma_{N}(M))$, denoted by $T(Gamma_{N,I}(M))$ is presented, where $I$ is an ideal of $R$. It is the graph with all elements of $...

Generalized Rings of Measurable and Continuous Functions

This paper is an attempt to generalize, simultaneously, the ring of real-valued continuous functions and the ring of real-valued measurable functions.

Generalized Derivations on Modules