Categories and Modules with K-Theory in View

Abstract

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Chain Conditions for Epsilon-Strongly Graded Rings with Applications to Leavitt Path Algebras

“…In other words, the category G-RING has arbitrary direct limits. The following lemma is a graded version of a well-known result (see [10,Prop. 5.2.14]).…”

“…A full matrix ring over R is von Neumann regular if and only if R is von Neumann regular. Moreover, recall that unital von Neumann regular rings are closed under direct limits (see[10, Prop. 5.2.14]).…”

Modules Over Color Hom-Poisson Algebras

“…The proof is similar to that of ( [14], Lemma 4.5). we recover the definition of Lie modules ( [15][16][17]).…”

Module theory over Leavitt path algebras and K-theory

“…We remark that B and U are adjoint functors (see [14,Proposition 3.3.15]). We know that B restricts to a functor between the categories of finitely presented modules and, by Proposition 5.9(1), the same applies to the subcategories of finite length modules.…”

“…Proof. Recall that two categories are equivalent if and only if there is a full, faithful and dense functor between them (see [14,Proposition 1.3.14]). By Proposition 6.2, the functor B satisfies the same natural property than T up to natural isomorphism, hence B is a category equivalence.…”