Direct product decomposition of commutative semi-simple rings

In this paper it is shown that a commutative semisimple ring is isomorphic to a direct product of fields if and only if it is hyperatomic and orthogonally complete. In this paper we give a necessary and sufficient condition for a commutative semisimple ring R (i.e., R has no nonzero nilpotent element) to be isomorphic to a direct product of fields. In particular, we show that hyperatomicity and orthogonal completeness is such a necessary and sufficient condition. It is well known that without these conditions R is isomorphic to a subring of a direct product of fields [1, p. 16]. We would like to emphasize that, in what follows, R stands for a commutative ring with no nonzero nilpotent element. Thus, in particular, for every element x of R, (1) X2 =O if and only if x = O. We first prove several lemmas. Lemma 1 below, generalizes the corresponding result for Boolean Rings [2, p. 154]. LEMMA 1. The ring R is partially ordered by : where for every element x and y of R, (2) x 5 y if and only if xy = x2. PROOF. Since xx=x2 it follows from (2) that x gx. Thus, ? is reflexive. Moreover, if x9y and y