The Semigroup of Ideals of a Fir Is

If R is a 2-sided fir (free ideal ring) with no non-trivial right invariant elements, we shall find that the non-zero 2-sided ideals of R, under the usual multiplication of ideals, form a free semigroup with 1. In particular, this holds when R is a free associative algebra over a field. (We also consider the operations of multiplying right ideals by 2-sided ideals to get right ideals, 2-sided ideals by left ideals to get left ideals, and right ideals by left ideals to get additive subsets, and find that these actions too are free, in the appropriate senses.) It is well known that the set V of proper subvarieties of the variety of associative algebras over a field K is in natural correspondence with the set T of non-zero T-ideals (completely invariant ideals) of any free /sf-algebra R = K(X} on an infinite set of indeterminates. A product of T-ideals is again a T-ideal, and we shall show that the non-zero T-ideals of such an algebra R form a free subsemigroup of the semigroup of all non-zero ideals. Thus the induced semigroup structure on V is also free. Finally, we show that in any 2-fir R without non-trivial invariant elements, a 2-sided ideal / is uniquely determined by its (/?, /?)-bimodule structure. (This result is independent of the others.)