Pseudobases in direct powers of an algebra

Author:
Paul Bankston

Journal:
Trans. Amer. Math. Soc. **335** (1993), 79-90

MSC:
Primary 08A35; Secondary 03C05, 12L10

DOI:
https://doi.org/10.1090/S0002-9947-1993-1155348-3

MathSciNet review:
1155348

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A subset of an abstract algebra is a *pseudobasis* if every function from into extends uniquely to an endomorphism on . is called -*free* has a pseudobasis of cardinality ; is *minimally free* if has a pseudobasis. (The 0-free algebras are "rigid" in the strong sense; the -free groups are always abelian, and are precisely the additive groups of -rings.) Our interest here is in the existence of pseudobases in direct powers of an algebra . On the positive side, if is a rigid division ring, is a cardinal, and there is no measurable cardinal with , then is -free whenever . On the negative side, if is a rigid division ring and there is a measurable cardinal with , then is not minimally free.

**[1]**P. Bankston and R. Schutt,*On minimally free algebras*, Canad. J. Math.**37**(1985), 963-978. MR**806650 (87b:08018)****[2]**P. Bankston,*A note on large minimally free algebras*, Algebra Universalis**26**(1989), 346-350. MR**1044854 (91b:08008)****[3]**-,*Minimal freeness and commutativity*, Algebra Universalis**29**(1992), 88-108. MR**1145558 (93d:08002)****[4]**P. Bankston and R. A. McCoy,*On the classification of minimally free rings of continuous functions*, General Topology and Applications (Proc. 1988 Northeast Conf., R. M. Shortt, ed.), Dekker, New York, 1990, pp. 51-58. MR**1057623 (91k:54027)****[5]**-, -*enrichments of topologies*, Topology Appl.**42**(1991), 37-55. MR**1135780 (92m:54002)****[6]**G. Birkhoff,*On the structure of abstract algebras*, Proc. Cambridge Philos. Soc.**31**(1935), 433-454.**[7]**C. C. Chang and H. J. Keisler,*Model theory*, North-Holland, Amsterdam, 1973.**[8]**W. W. Comfort and S. Negrepontis,*The theory of ultrafilters*, Springer-Verlag, Berlin, 1974. MR**0396267 (53:135)****[9]**M. Dugas, A. Mader, and C. Vinsohaler,*Large*-*rings exist*, J. Algebra**108**(1987), 88-101. MR**887193 (88e:16047)****[10]**E. Fried and J. Sichler,*Homomorphisms of commutative rings with unit elements*, Pacific J. Math.**45**(1973), 485-491. MR**0335489 (49:270)****[11]**G. Grätzer,*Universal algebra*, 2nd ed., Springer-Verlag, New York, 1979. MR**538623 (80g:08001)****[12]**I. Kříž and A. Pultr,*Large*-*free algebras*, Algebra Universalis**21**(1985), 46-53. MR**835969 (87f:08005)****[13]**P. Pröhle,*Does a given subfield of characteristic zero imply any restriction to the endomorphism monoids of fields*? Acta Math.**50**(1986), 15-38. MR**862178 (88k:12009)****[14]**R. Schutt, (private communication).

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
08A35,
03C05,
12L10

Retrieve articles in all journals with MSC: 08A35, 03C05, 12L10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1993-1155348-3

Keywords:
Pseudobases,
direct powers,
universal algebra,
minimally free,
fields,
division rings,
measurable cardinals

Article copyright:
© Copyright 1993
American Mathematical Society