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In this paper, we compare $(n,m)$-purities for different pairs of positive integers $(n,m)$. When $R$ is a commutative ring, these purities are not equivalent if $R$ doesnt satisfy the following property: there exists a positive integer $p$ such that, for each maximal ideal $P$, every finitely generated ideal of $R_P$ is $p$-generated. When this property holds, then the $(n,m)$-purity and the $(n,m)$-purity are equivalent if $m$ and $m$ are integers $geq np$. These results are obtained by a generalization of Warfields methods. There are also some interesting results when $R$ is a semiperfect strongly $pi$-regular ring. We also compare $(n,m)$-flatnesses and $(n,m)$-injectivities for different pairs of positive integers $(n,m)$. In particular, if $R$ is right perfect and right self $(aleph_0,1)$-injective, then each $(1,1)$-flat right $R$-module is projective. In several cases, for each positive integer $p$, all $(n,p)$-flatnesses are equivalent. But there are some examples where the $(1,p)$-flatness is not equivalent to the $(1,p+1)$-flatness.
The aim of this note is to describe the structure of finite meadows. We will show that the class of finite meadows is the closure of the class of finite fields under finite products. As a corollary, we obtain a unique representation of minimal meadows in terms of prime fields.
In this paper we analyzed solutions of some complex matrix equations related to pseudoinverses using the concept of reproductivity. Especially for matrix equation AXB=C it is shown that Penroses general solution is actually the case of the reproductive solution.
This short survey contains some recent developments of the algebraic theory of racks and quandles. We report on some elements of representation theory of quandles and ring theoretic approach to quandles.
The Hochschild cohomology of a tensor product of algebras is isomorphic to a graded tensor product of Hochschild cohomology algebras, as a Gerstenhaber algebra. A similar result holds when the tensor product is twisted by a bicharacter. We present ne
In this paper we prove that any local automorphism on the solvable Leibniz algebras with null-filiform and naturally graded non-Lie filiform nilradicals, whose dimension of complementary space is maximal is an automorphism. Furthermore, the same prob