Chain conditions on symmetric elements

Author:
Susan Montgomery

Journal:
Proc. Amer. Math. Soc. **46** (1974), 325-331

MSC:
Primary 16A28

MathSciNet review:
0349736

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Abstract: Recently Britten has proven an analog of Goldie's theorem for the Jordan ring of symmetric elements in a ring with involution of characteristic not 2. In this paper we first extend Britten's theorem to the situation where is an arbitrary ring and the Jordan ring is only an ample subring of the symmetric elements. We apply this result to show that if has ACC on quadratic ideals, then the (Jordan) nil radical of is nilpotent.

**[1]**Daniel J. Britten,*On prime Jordan rings 𝐻(𝑅) with chain condition*, J. Algebra**27**(1973), 414–421. MR**0325714****[2]**T. S. Erickson and S. Montgomery,*The prime radical in special Jordan rings*, Trans. Amer. Math. Soc.**156**(1971), 155–164. MR**0274543**, 10.1090/S0002-9947-1971-0274543-4**[3]**I. N. Herstein,*Topics in ring theory*, The University of Chicago Press, Chicago, Ill.-London, 1969. MR**0271135****[4]**Charles Lanski,*Nil subrings of Goldie rings are nilpotent*, Canad. J. Math.**21**(1969), 904–907. MR**0248174****[5]**Charles Lanski,*Chain conditions in rings with involution*, J. London Math. Soc. (2)**9**(1974/75), 93–102. MR**0360676****[6]**Kevin McCrimmon,*On Herstein’s theorems relating Jordan and associative algebras*, J. Algebra**13**(1969), 382–392. MR**0249476****[7]**Susan Montgomery,*Rings of quotients for a class of special Jordan rings*, J. Algebra**31**(1974), 154–165. MR**0374203**

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1974-0349736-6

Keywords:
Involution,
symmetric elements,
Goldie ring,
nil radical

Article copyright:
© Copyright 1974
American Mathematical Society