Simple near-ring centralizers of finite rings

Abstract

For a finite ring <italic>R</italic> with identity and a finite unital <italic>R</italic>-module <italic>V</italic> we call <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C left-parenthesis upper R right-parenthesis equals left-brace f colon upper V right-arrow upper V vertical-bar f left-parenthesis alpha v right-parenthesis equals alpha f left-parenthesis v right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>R</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mi>V</mml:mi> <mml:mo stretchy="false">→<!-- → --></mml:mo> <mml:mi>V</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>α<!-- α --></mml:mi> <mml:mi>v</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>α<!-- α --></mml:mi> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>v</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C(R) = \{ f:V \to V|f(\alpha v) = \alpha f(v)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha element-of upper R comma v element-of upper V right-brace"> <mml:semantics> <mml:mrow> <mml:mi>α<!-- α --></mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:mi>v</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi>V</mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\alpha \in R,v \in V\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the near-ring centralizer of <italic>R</italic>. We investigate the structure of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C left-parenthesis upper R right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>R</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C(R)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and obtain a characterization of those rings <italic>R</italic> for which <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C left-parenthesis upper R right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>R</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C(R)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a simple nonring.

Locations

• Proceedings of the American Mathematical Society - View - PDF