Embeddings into simple associative algebras

INTRODUCTION In a recent paper [i], Hall showed that if Kt ..... K~ are any nontrivial groups and the group / satisfies the condition I/,J6 iK.*..~ [, then can be embedded in a simple group $ conraining s ~# and generated by them: $ = ~/ .... ,~,> We will con~ider the analogous question for associative rings and algebras. Let us first mention that in the [2] it was proved that any associative algebra can be embedded in a simple associative algebra. Regarding associative rings, a ring ~ can be embedded in a simple ring if and only if it has a characteristic, i.e., either its additive group is torsion-free or there exists a prime such that /~,.r= O,~g:~. In the first case, ~ can be embedded in an algebra over the field of rational numbers; in the second, R is an algebra over the field with p elements. Thus the problem for rings reduces to the problem for algebras over an a~ most countable field. THEOREM I. Suppose [ is an at most countable field and A , and ~ is a simple ring, then all four rings ~. <./(2,Kj have the same characteristic ~ ~ @ . In the general case we have THEOREM i'. Suppose ~ is any field and A,~,~2, 6 are nonzero associative algebras over ~ such that I / l l .~ l~ l~6 ~ }(3 I and ~fTL~ I~ ~(2 4~ ~ ~ l~I . Then A canbe embedded in a simple associative algebra ~ generated by subalgebras /(i'/(2,/(3" Thus, in Theorem i' there is one additional restriction (~Lrrb~ ~/~2 ~ ?(3 ~ ]~I~, which, evidently, is also necessary. COROLLAKY. Any countable associative algebra can be embedded in a simple associative algebra with three generators. We now consider the question of when an algebra ~ is a sum of subalgebras ~. In this case we consider, instead of three, four subalgebras 7(I'''" A/, of infinite dimension over In order to formulate the theorem we require a certain condition on the algebras ~. Definition. Suppose ~ is an associative algebra over a field ~. ~qI~ =OG ~ ~ 9 We say that ~ satisfies condition (*) if ~( contains a countable series