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Download Advances in Ring Theory by Toma Albu, Robert Wisbauer (auth.), S. K. Jain, S. Tariq PDF

By Toma Albu, Robert Wisbauer (auth.), S. K. Jain, S. Tariq Rizvi (eds.)

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Colchester 1993 (editor: P. Higgins), Vniv. Essex 1994, pp. 60-69. 10. , Congruence Compact Acts, Semigroup Forum, to appear. 11. , Duality Theory for Grothendieck Categories and Linearly Compact Rings, J. Algebra 15 (1970), 473-542. 12. , On Subdirect Sums of Simple Rings with Unity, Bull. Acad. Polon. Sci. 8 (1960), 223-228. 13. , The Brown-McCoy Radical in Categories, Fund. Math. 59 (1966), 23-4l. 14. sZ, F. , Radicals of Rings, John Wiley & Sons, 1981. 15. Wiegandt, R, Radical and Semisimplicity in Categories, Acta Math.

If R satisfies any of the following conditions, then R = J(R) EB M(R) (ring direct sum) and J(R) = Z2(RR) = Z2(RR): (i) R is an FI ring. (ii) R is semiprime. (iii) Every right semicentral idempotent is central. (iv) Every left semicentral idempotent is central. (v) R is GFC. (vi) R = EBf=l eiR, where {ei I i = 1,2, ... , n} is a complete set of orthogonal primitive idempotents. (vii) Soc(RR) is left essential in Rand SOC(RR) is right essential in R. F. Y. K. PARK Proof. 1. 4, parts (ii), (iii), (iv) and (v) imply that R is an FI ring.

Jain et al. M5-1EX 50 ROBERT A. BROWN AND MARY H. 1. A:::; M is small in M , denoted A ~ M, provided that, for each B :::; M, A + B = M implies that B = M. M is hollow if every proper submodule of M is small. 2. Let A :::; M. B is a supplement of A in M provided that B is minimal among submodules C of M such that A + C = M. 3. Suppose that B + Be = M. Then Be is a supplement of B o in M if and only if B n Be ~ Be. Proof. ~: Suppose that B n Be + C = Be. Then M = B + Be = B + B n Be + C = B + C.

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