Li, Lide; Schein, Boris M. Strongly regular rings. (English) Zbl 0571.16008 Semigroup Forum 32, 145-161 (1985). A ring is called strongly regular if its multiplicative semigroup is inverse. This definition is equivalent to more conventional definitions of strongly regular rings [for example, to a definition given by R. Arens and I. Kaplansky, Trans. Am. Math. Soc. 63, 457-481 (1948; Zbl 0032.00702)]. The authors rely heavily on a previous article by the second author which contains various equivalent characterizations of strongly regular rings [Izv. Vyssh. Uchebn. Zaved. Mat. 1966, No. 2(51), 111-122 (1966; Zbl 0208.298)]. This article contains three theorems. Theorem 1. A regular ring is not strongly regular if and only if it contains an isomorphic copy of the ring of all \(2\times 2\) matrices over a prime field. It follows that the smallest regular but not strongly regular ring has order 16 and is isomorphic to \(Z(2)_ 2\), the ring of \(2\times 2\) matrices over \(Z(2)\), the field of order 2. – Theorem 2. A semigroup identity is satisfied by idempotents of a regular ring which is not strongly regular if and only if it is satisfied by idempotents of \(Z(2)_ 2\). Let \(u=v\) be a semigroup identity. The initial part of \(u\) is the word obtained from \(u\) by deleting all non-first occurrences of all variables in \(u\). The identity is called coinitial if the initial parts of \(u\) and \(v\) coincide. It is called cofinal if the final parts of \(u\) and \(v\) coincide. Final parts are defined analogously to initial parts. If \(a\) and \(b\) are semigroup words, we write \(a\leq b\) if \(a\) and \(b\) have the same first letter, the same last letter, and each letter occurring in \(a\) occurs in \(b\) as well. If \(a\), \(b\), and \(c\) are (possibly empty) words, then \(b\) is called a subword of the word \(abc\). A segment of a word \(u\) is a subword \(w\) of \(u\) such that the first and last letters of \(w\) are different and both of them occur in \(w\) only once. An identity \(u=v\) is called smooth if, for every segment \(w\) of \(u\) there exists a segment \(w'\) of \(v\) such that \(w'\leq w\), and for every segment \(w'\) of \(v\) there exists a segment \(w\) of \(u\) such that \(w\leq w'\). Theorem 3. A semigroup identity is satisfied by idempotents of a regular ring which is not strongly regular if and only if it is coinitial, cofinal, and smooth. Thus one obtains an effective algorithm which determines whether an identity satisfied by idempotents of a regular ring forces the ring to be strongly regular. Cited in 1 Document MSC: 16E50 von Neumann regular rings and generalizations (associative algebraic aspects) 20M05 Free semigroups, generators and relations, word problems 20M25 Semigroup rings, multiplicative semigroups of rings 16U99 Conditions on elements 16S50 Endomorphism rings; matrix rings 16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) Keywords:multiplicative semigroup; strongly regular rings; ring of \(2\times 2\) matrices; semigroup identity; idempotents; initial parts; semigroup words Citations:Zbl 0032.00702; Zbl 0208.298 PDFBibTeX XMLCite \textit{L. Li} and \textit{B. M. Schein}, Semigroup Forum 32, 145--161 (1985; Zbl 0571.16008) Full Text: DOI EuDML References: [1] Arens, R. and I. Kaplansky,Topological representation of algebras, Trans. Amer. Math. Soc. 63(1948), 457–481. · Zbl 0032.00702 · doi:10.1090/S0002-9947-1948-0025453-6 [2] Schein, B.M., O-rings and LA-rings, Izvestiya Vysshikh Uchebnykh Zavedenii, Matematika, 1965, no. 2, 111–122 [Russian; English translation in Amer. Math. Soc. Translations (2) 96(1970), 137–152.] This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.