#### Document Type

Article

#### Language

eng

#### Format of Original

12 p.

#### Publication Date

1-1993

#### Publisher

American Mathematical Society

#### Source Publication

Transactions of the American Mathematical Society

#### Source ISSN

0002-9947

#### Original Item ID

doi: 10.1090/S0002-9947-1993-1155348-3; Shelves: QA1 .A522 Storage S

#### Abstract

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 *A ^{I}* 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

*A*is

^{I}*K*-free whenever . On the negative side, if

*A*is a rigid division ring and there is a measurable cardinal

*u*with ,then

*A*is not minimally free.

^{I}
## Comments

Published version.

Transactions of the American Mathematical Society, Vol. 335, No. 1 (January 1993): 79-90. DOI. © The American Mathematical Society 1993. Used with permission.