Solvable assosymmetric rings are nilpotent

Abstract

Assosymmetric rings are ones which satisfy the law <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis x comma y comma z right-parenthesis equals left-parenthesis upper P left-parenthesis x right-parenthesis comma upper P left-parenthesis y right-parenthesis comma upper P left-parenthesis z right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>P</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>P</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>P</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(x,y,z) = (P(x),P(y),P(z))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for each permutation <italic>P</italic> of <italic>x, y, z</italic> . Let <italic>A</italic> be an assosymmetric ring having characteristic different from 2 or 3. We show that if <italic>A</italic> is solvable then <italic>A</italic> is nilpotent. Also, if each subring generated by a single element is nilpotent, and if <italic>A</italic> has D.C.C. on right ideals, then <italic>A</italic> is nilpotent. We also give an example showing that the Wedderburn Principal Theorem fails for assosymmetirc rings.