On rings with zero total
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.
scientific article; zbMATH DE number 1066095