An Extension of the Jacobson Radical

Prime ideals in nonassociative rings

“…A ring R is primitive if it contains a modular maximal right ideal M which contains no nonzero ideal of F. We say that an ideal A of R is primitive if the ring R/A is primitive. Brown [5,Theorem l] has shown that J(R) is an intersection of primitive ideals, and we shall show below, as a corollary to the next theorem, that a primitive ideal is actually re-prime for each u £21. The proof is by induction on the degree of u, and we begin by considering the special case in which re = XiX2.…”

“…The Jacobson radical. The definition of the Jacobson radical of an associative ring has been extended by Brown [5] to the nonassociative case. The present treatment is in terms of the right radical, which for nonassociative rings need not coincide with the left radical.…”

“…Under the assumption that v(a) is quasi-regular, we thus have a -cEQ(a -c). A lemma of Brown [5] shows that Q(a -c)C.Q(a)+(c)r, where (c)r is the right ideal in R generated by c. Since c£(? (o), we have (c)rQQ(a), so finally aEQ(a).…”

“…Following Brown [5], we say that a right ideal I in R is modular if there exists an element e of R such that er -rEI for all rER. A ring R is primitive if it contains a modular maximal right ideal M which contains no nonzero ideal of F. We say that an ideal A of R is primitive if the ring R/A is primitive.…”

“…The radical defined by Jacobson for an associative ring has been generalized by Brown [5] to the nonassociative case. If / is this radical of the ring R, we show in §6 that / is i/-prime for each zj£93 and, more precisely, that a primitive ideal is itself M-prime for each wGSI-In the final section we briefly indicate the relation of the results of this paper to a radical studied by Smiley [9].…”