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]**William C. Brown,*Strong inertial coefficient rings*, Michigan Math. J.**17**(1970), 73–84. MR**0263802****[2]**W. C. Brown and E. C. Ingraham,*A characterization of semilocal inertial coefficient rings*, Proc. Amer. Math. Soc.**26**(1970), 10–14. MR**0260730**, https://doi.org/10.1090/S0002-9939-1970-0260730-2**[3]**W. C. Brown,*Some splitting theorems for algebras over commutative rings*, Trans. Amer. Math. Soc.**162**(1971), 303–315. MR**0284428**, https://doi.org/10.1090/S0002-9947-1971-0284428-5**[4]**Charles W. Curtis and Irving Reiner,*Representation theory of finite groups and associative algebras*, Pure and Applied Mathematics, Vol. XI, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1962. MR**0144979****[5]**A. R. Magid,*Pierce’s representation and separable algebras*, Illinois J. Math.**15**(1971), 114–121. MR**0272832****[6]**R. S. Pierce,*Modules over commutative regular rings*, Memoirs of the American Mathematical Society, No. 70, American Mathematical Society, Providence, R.I., 1967. MR**0217056**

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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