Decompositions of Abelian -groups

Author:
R. W. Stringall

Journal:
Proc. Amer. Math. Soc. **28** (1971), 409-410

MSC:
Primary 20.30

DOI:
https://doi.org/10.1090/S0002-9939-1971-0274582-9

MathSciNet review:
0274582

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Using some elementary properties of endomorphism rings and their radical ideals, an equivalence between the category of -rings and the category of Boolean rings and some examples introduced by the author, it is shown that for every countable atomic Boolean algebra there is a -group without elements of infinite height, standard basic subgroup and no proper isomorphic subgroups which contains a maximal lattice of summands isomorphic to the given Boolean algebra. Moreover, it is established that this lattice is representative in the sense that it determines, up to isomorphism, all the summands of the group.

**[1]**N. Jacobson,*Structure of rings*, Amer. Math. Soc. Colloq. Publ., vol. 37, Amer. Math. Soc., Providence, R. I., 1956. MR**18**, 373. MR**0081264 (18:373d)****[2]**N. H. McCoy and D. Montgomery,*A representation of generalized Boolean rings*, Duke Math. J.**3**(1937), 455-459. MR**1546001****[3]**R. Sikorski,*Boolean algebras*, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, Heft 25, Springer-Verlag, New York, 1969. MR**39**#4053. MR**0242724 (39:4053)****[4]**R. W. Stringall,*Endomorphism rings of primary Abelian groups*, Pacific J. Math.**20**(1967), 535-557. MR**34**#7644. MR**0207830 (34:7644)****[5]**-,*The categories of -rings are equivalent*, Proc. Amer. Math. Soc. (to appear).

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
20.30

Retrieve articles in all journals with MSC: 20.30

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1971-0274582-9

Keywords:
Boolean algebra of idempotents,
lattice of summands,
-groups without proper isomorphic subgroups,
endomorphism ring,
Jacobson radical,
-ring,
subdirect product of finite fields

Article copyright:
© Copyright 1971
American Mathematical Society