Format of Original
American Mathematical Society
Transactions of the American Mathematical Society
Original Item ID
doi: 10.1090/S0002-9947-1993-1155348-3; Shelves: QA1 .A522 Storage S
A subset P of an abstract algebra A is a pseudobasis if every function from P into A extends uniquely to an endomorphism on A. A is called K-free has a pseudobasis of cardinality K; A is minimally free if A has a pseudobasis. (The 0-free algebras are "rigid" in the strong sense; the 1-free groups are always abelian, and are precisely the additive groups of E-rings.) Our interest here is in the existence of pseudobases in direct powers AI of an algebra A. On the positive side, if A is a rigid division ring, K is a cardinal, and there is no measurable cardinal u with , then AI is K-free whenever . On the negative side, if A is a rigid division ring and there is a measurable cardinal u with ,then AI is not minimally free.