On subdirect decomposition and varieties of some rings with involution (Q1856591)

A well-known result of N. Jacobson says that every ring satisfying the identity \(x^{n+1}=x\) for some \(n\geq 1\) is commutative, and is in fact a subdirect product of finite fields. The purpose of this paper is to characterize rings with involution which satisfy this identity. The finite field of order \(p^k\) with the identity involution will be denoted \(\text{GF}(p^k)\) and the same field with involution \(*\) defined by \(x^*:=x^{p^m}\) where \(k=2m\) will be written \(\text{GF}^*(p^k)\). If \(R\) is a ring, then the ring \(R\oplus R^{op}\) with the exchange involution is denoted \(\text{Ex}(R)\). It is shown that a ring with involution \(R\) is subdirectly irreducible and satisfies the identity \(x^{n+1}=x\) if and only if there exits a prime \(p\) and a positive integer \(k\) such that \((p^k-1)\mid n\) and \(R\) is isomorphic to one of \(\text{GF}(p^k)\), \(\text{GF}^*(p^k)\) or \(\text{Ex}(\text{GF}(p^k))\). The lattice of all varieties of rings with involution satisfying \(x^{n+1}=x\) is then considered. Let \(L\) denote this lattice and let \(\{p_1,\dots,p_k\}\) be the set of all primes such that \((p_i-1)\mid n\), \(1\leq i\leq k\). Let \(L_{p_i}\) be the sublattice of \(L\) consisting of varieties of characteristic \(p_i\). Then \(L\cong L_{p_1}\times\cdots\times L_{p_k}\). The theory is used to give a full description of the lattice of all varieties of rings with involution satisfying \(x^7=x\).