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 \textit{R. Arens} and \textit{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.