Central semicommutative rings

L. Wang () and J. C. Wei ()
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L. Wang: Southeast University
J. C. Wei: Yangzhou University

Indian Journal of Pure and Applied Mathematics, 2014, vol. 45, issue 1, 13-26

Abstract: Abstract A ring R is central semicommutative if ab = 0 implies that aRb ⊆ Z(R) for any a, b ∈ R. Since every semicommutative ring is central semicommutative, we study sufficient condition for central semicommutative rings to be semicommutative. We prove that some results of semicommutative rings can be extended to central semicommutative rings for this general settings, in particular, it is shown that every central semicommutative ring is nil-semicommutative. We show that the class of central semicommutative rings lies strictly between classes of semicommutative rings and abelian rings. For an Armendariz ring R, we prove that R is central semicommutative if and only if the polynomial ring R[x] is central semicommutative. Moreover, for a central semicommutative ring R, it is proven that (1) R is strongly regular if and only if R is a left GP-V′-ring whose maximal essential left ideals are GW-ideals if and only if R is a left GP-V′-ring whose maximal essential right ideals are GW-ideals. (2) If R is a left SF and central semicommutative ring, then R is a strongly regular ring.

Keywords: Nil-semicommutative ring; GP-injective; strongly regular ring; SF ring (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/s13226-014-0048-9

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