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In this paper we improve recent results dealing with cellular covers of R-modules. Cellular covers (sometimes called colocalizations) come up in the context of homotopical localization of topological spaces. They are related to idempotent cotriples, idempotent comonads or coreflectors in category theory.
Recall that a homomorphism of R-modules π: G → H is called a cellular cover over H if π induces an isomorphism , where π⁎(φ) = πφ for each (where maps are acting on the left). On the one hand,...
Let F be a field, and let R be a finitely-generated F-algebra, which is a domain with quadratic growth. It is shown that either the center of R is a finitely-generated F-algebra or R satisfies a polynomial identity (is PI) or else R is algebraic over F. Let r ∈ R be not algebraic over F and let C be the centralizer of r. It is shown that either the quotient ring of C is a finitely-generated division algebra of Gelfand-Kirillov dimension 1 or R is PI.
The purpose of this paper is to investigate identities satisfied by centralizers on prime and semiprime rings. We prove the following result: Let be a noncommutative prime ring of characteristic different from two and let and be left centralizers on . Suppose that is fulfilled for all . If
The main result: Let be a -torsion free semiprime ring and let be an additive mapping. Suppose that holds for all . In this case is a centralizer.
Let be a commutative ring, be a generalized matrix algebra over with weakly loyal bimodule and be the center of . Suppose that is an -bilinear mapping and that is a trace of . The aim of this article is to describe the form of satisfying the centralizing condition (and commuting condition ) for all . More precisely, we will revisit the question of when the centralizing trace (and commuting trace) has the so-called proper form from a new perspective. Using the aforementioned...
We determine when an element in a noncommutative ring is the sum of an idempotent and a radical element that commute. We prove that a matrix over a projective-free ring is strongly -clean if and only if , or , or is similar to , where , , and the equation has a root in and a root in . We further prove that is strongly -clean if be optimally -clean.
A ring is (weakly) nil clean provided that every element in is the sum of a (weak) idempotent and a nilpotent. We characterize nil and weakly nil matrix rings over abelian rings. Let be abelian, and let . We prove that is nil clean if and only if is Boolean and is nil. Furthermore, we prove that is weakly nil clean if and only if is periodic; is , or where is a Boolean ring, and that is weakly nil clean if and only if is nil clean for all .
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