Boolean rings (Q1078218)

The two theorems of this paper provide new characterizations of Boolean rings, while the five lemmas have also an intrinsic interest; thus e.g. Lemma 3 is a strengthening of a theorem due to \textit{C. E. Rickart} [Bull. Am. Math. Soc. 54, 758-764 (1948; Zbl 0032.249)]. We reproduce below Theorem 1, which sharpens a result of \textit{G. Thierrin} [Simon Stevin 55, 37-40 (1981; Zbl 0459.16014)]: Let R be an arbitrary ring and for each \(a\in R\) set \(\ell (x)=\{z\in R|\) \(zx=0\}\) and \(r(x)=\{z\in R|\) \(xz=0\}\). Then the following conditions are equivalent: (1) R is a Boolean ring; (2) \(x\neq y\Rightarrow \ell (x)\neq \ell (y)\) (or equivalently r(x)\(\neq r(y))\); (3) \((x^ n=0\Rightarrow x=0)\) and \((\ell (x)=\ell (y) \& r(x)=r(y)\Rightarrow x=y)\).