Additive Groups of Rings (Chapman & Hall CRC Research Notes - download pdf or read online

By Shalom Feigelstock

ISBN-10: 0273085913

ISBN-13: 9780273085911

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Extra resources for Additive Groups of Rings (Chapman & Hall CRC Research Notes in Mathematics Series)

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2.. 6 remains true if the word "strongly" is deleted. 7: Let G be a torsion free strongly principal ideal ring group. Then G is strongly indecomposable. 6. 8: Let G be a torsion group. The following are equivalent: G is bounded. G is an associative principal ideal ring group. G is a principal ideal ring group. 1) 2) 3) Proof: 1) • 2): Suppose that nG = 0, n a positive integer. 9. 3]. The ring direct sum R = k(£) R k is an associative principal ideal ring satisfying P In P and R2 '} 0. 2) 46 ~ 3): Obvious.

5. If n = ""• then G is nil, Proposition 1. 1, a contradiction. Hence H is cyclic. 1, a contradict1on. 2) Suppose that G is mixed. 5, and so G = H<±> K, H 'I 0, K 'I 0. 2. 11, both H and K are nil. 1 it may be assumed that H is a torsion group, and that K is torsion free. Let R be a Noetherian ring with R+ = G. Since ti <2 R, H = . Let n = k rr lhil. i =1 Then nH = 0. 1, H is divisible, a contradiction. 13: Let G be a strongly Noetherian ring group. and G/Gt are strongly Noetherian ring groups.

1) G is a field group. 2) G is a division ring group. 3) G is a simple ring group The following are 4) G is divisible. e •• G ~ (i) Q+. 3). Proof: The implications 1) • 2) • 3) are obvious. 3) • 4): Let R be a simple ring with R+ =G. Then nR is an ideal in R for every nonzero integer n. Hence (nR)+ = nG = G. and so G is divisible. 4) • 1): Let G = ffi Q+. oc an arbitrary cardinal. Then G is the oc additive group of a field extension of degree oc of the field of rational numbers. 8]): Let {qi li E IJ be a set of primes.

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Additive Groups of Rings (Chapman & Hall CRC Research Notes in Mathematics Series) by Shalom Feigelstock


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