On rings with zero total. (English) Zbl 0884.16006

The total of a ring \(A\) is the subset \(\text{Tot}(A)=\{a\in A:aA\) does not contain nonzero idempotents}. Let \(\text{In}(a)=\min(n:a^n=0)\) for a nilpotent element \(a\in A\) and \(\text{In}(A)=\sup(\text{In}(a):a\) nilpotent in \(A)\). For a ring \(A\) with \(\text{In}(A)=n<\infty\) the following are equivalent: i) \(\text{Tot}(A)=0\), ii) \(A\) contains an essential ideal \(I\) which is a direct sum of ideals \(I_k=M_{n_k}(D_k)\), \(k=1,2,\dots,t\), where \(n_1<n_2<\cdots<n_t=n\), each \(M_{n_k}(D_k)\) is a matrix ring over a reduced ring \(D_k\), and if \(L\neq 0\) is a right ideal of \(D_k\), then the set of all central idempotents of \(D_k\) belonging to \(L\) generates an essential ideal in \(L\). This result yields interesting corollaries, e.g. a ring \(A\) with \(\text{In}(A)=n<\infty\) is a prime ring and \(\text{Tot}(A)=0\) if and only if \(A\cong M_n(D)\), where \(D\) is a division ring; the maximal right ring of quotients of \(A\) equals the maximal left ring of quotients and is isomorphic to a finite direct sum of matrix rings over Abelian regular selfinjective rings.


16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
16S90 Torsion theories; radicals on module categories (associative algebraic aspects)
16N40 Nil and nilpotent radicals, sets, ideals, associative rings
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
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