A splitting theorem for algebras over commutative von Neumann regular rings

Author:
William C. Brown

Journal:
Proc. Amer. Math. Soc. **36** (1972), 369-374

MSC:
Primary 16A16; Secondary 16A56

DOI:
https://doi.org/10.1090/S0002-9939-1972-0314887-7

MathSciNet review:
0314887

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Abstract: Let *R* be a commutative von Neumann ring. Let *A* be an *R*-algebra which is finitely generated as an *R*-module and has separable over *R*. Here *N* is the Jacobson radical of *A*. Then it is shown that there exists an *R*-separable subalgebra *S* of *A* such that and . Further it is shown that if *T* is another *R*-separable subalgebra of *A* for which and , then there exists an element such that . This result is then used to determine the structure of all strong inertial coefficient rings.

**[1]**W. C. Brown,*Strong inertial coefficient rings*, Michigan Math. J.**17**(1970), 73-84. MR**41**#8402. MR**0263802 (41:8402)****[2]**W. C. Brown and E. C. Ingraham,*A characterization of semilocal inertial coefficient rings*, Proc. Amer. Math. Soc.**26**(1970), 10-14. MR**41**#5354. MR**0260730 (41:5354)****[3]**W. C. Brown,*Some splitting theorems for algebras over commutative rings*, Trans. Amer. Math. Soc.**162**(1971), 303-315. MR**0284428 (44:1655)****[4]**C. W. Curtiss and I. Reiner,*Representation theory of finite groups and associative algebras*, Interscience, New York, 1962. MR**26**#2519. MR**0144979 (26:2519)****[5]**A. R. Magid,*Pierce's representation and separable algebras*, Illinois J. Math.**15**(1971), 114-121. MR**42**#7713. MR**0272832 (42:7713)****[6]**R. S. Pierce,*Modules over commutative regular rings*, Mem. Amer. Math. Soc. No. 70 (1967). MR**36**#151. MR**0217056 (36:151)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1972-0314887-7

Keywords:
von Neumann ring,
strong inertial coefficient ring

Article copyright:
© Copyright 1972
American Mathematical Society