By Shalom Feigelstock
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Additional info for Additive Groups of Rings (Chapman & Hall CRC Research Notes in Mathematics Series)
The following are equivalent: 1) G is the additive group of an (associative) ring satisfying the DCC for ideals. n ni 2) G ... ) (3 a prime. m, n; a i=l j=l aj non-negative integers. a. aj arbitrary cardinals. i = 1, ••• ,m; 56 j = 1 , ••• ,ni . 13: Let G be a group. 10. 14: 1) 2) Let G be a non-nil group. The following are equivalent: Every ring R with R+ = G, and R2 1 0 satisfies the DCC for ideals. m ni . G ""c±)Q+ (±) (±) (±) Z(p~) 0 Z(p~), p. a prime, m, ni 1 1 1 a i=l j=l a. • ,ni. 15: Let R be an (associative) ring with trivial annihilator, such that R+ = D<+l (R+)t• D a divisible group.
3: Let G be a torsion free group with R+ = G. be a nil associative ring with r(G) = n, and let R Then Rn+l = 0. Proof: Let 0 ~ x € R, and let m be the smallest positive integer such that xm = 0. Suppose that m > n+l. , i =1 am-lx m-1 m-2 - - a xm-1 = m-1 i aix . I: i =1 If a1 = 0, 1 = 1 , ••• ,m-2, then am- 1 ~ 0, and o. Since G is torsi on free xm-1 = 0, a contradiction. Therefore a; ~ 0 for some . , a xmm-2 = ~ i ~ m-2. x m-1 i=l 1 m-3 . x 1 + • Repeating the above procedure m-3 more time . 1 1= 1 yields that a1xm-l = 0, with a 1 ~a.
The ring direct sum S = F(±) R is associative. possesses only finitely many ideals. and s+ ~G. 11: Let G be a non-nil group. The following are equivalent: 1) Every (associative) ring R with R+ = G, and R2 '! 0 possesses only finitely many ideals. n. + m 1 . ,9 Q (±} (t) l+l Z(p~). pi a prime. m, ni non-negative integers. j=l a. J a. 12: Let G be a group. The following are equivalent: 1) G is the additive group of an (associative) ring satisfying the DCC for ideals. n ni 2) G ... ) (3 a prime. m, n; a i=l j=l aj non-negative integers.
Additive Groups of Rings (Chapman & Hall CRC Research Notes in Mathematics Series) by Shalom Feigelstock