A note on Baer rings
All rings in this paper are commutative rings with unit. For an element a of a ring R denote by \((a)^{\perp}\) the annihilator of the principal ideal (a). If \((a)^{\perp}=(e)\) for some idempotent e of R then e is unique and \(a^ 0\) denotes the element 1-e. A ring R is said to be a Baer ring if for every \(a\in R\) there exists an idempotent \(e\in R\) such that \((a)=(e)\). A homomorphism \(h: R\to R'\) is said to be R-compatible if whenever \((a)^{\perp}=(b)^{\perp}\) in R then \((h(a))^{\perp}=(h(b))^{\perp}\) in R'. If R and R' are Baer rings than an R-compatible homomorphism \(h: R\to R'\) is termed Baer homomorphism. In a previous paper [Ann. Univ. Ferrara, Nuova Ser., Sez. VII 25, 219-224 (1979; Zbl 0445.13003)], the author constructed an embedding of an arbitrary reduced ring R into a Baer ring \(R^ B\) by an R-compatible ring homomorphism \(i: R\to R^ B.\) The statement that the embedding is universal, i.e. for every Baer ring S and every R-compatible homomorphism \(h: R\to S\) there is an induced Baer homomorphism \(h^{\#}:\quad R^ B\to S\) such that \(h^{\#}(i(r))=h(r)\) and \(h^{\#}(i(r)^ 0)=h(r)^ 0\) for all \(r\in R\), was not correct. The aim of the present paper is to correct that result. The author gives a necessary and sufficient condition on R for \(i: R\to R^ B\) to be universal (theorem 2.2 and 2.14). Moreover the author constructs a reduced ring R which fails to satisfy the condition of the theorems.
scientific article; zbMATH DE number 4065134