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- W2132871052 abstract "Let A, B be rings and P a radical property. Call B an A-Algebra if B is an A-bimodule such that (ba)b1 = b(ab1), (bb1)a = b(b1a), a(bb1) = (ab)b1 for any a ∈ A and any b,b1 ∈ B. A ring R, written as R = A ⨄ B, is called a quasi-direct sum of (A, B) if A is a subring of R, B is an ideal of R and R is a direct sum of A and B as additive groups. The following results are obtained: 1. A quasi-direct sum of (A, B) is uniquely determined by an A-Algebra B (up to isomorphism); 2. The P-radical of the Algebra B is the same as the P-radical of the ring B; 3. P(A ⨄ B) = P(A) +(B) if and only if P(A)B + BP(A) ⊆ P(B); 4. If B has an identity e then P(A ⨄ B) = P(A)(1−e) + P(B); 5. If P(Z) = 0 for the integer ring Z, then P(Mn(R)) = Mn(P(R)) holds for all rings R if and only if the above equality holds for all unitary rings R. In addition, some relationships of radicals between rings (or algebras over a field, semigroup algebras, etc.) and their corresponding identity extensions are discussed." @default.
- W2132871052 created "2016-06-24" @default.
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- W2132871052 date "1997-01-01" @default.
- W2132871052 modified "2023-09-25" @default.
- W2132871052 title "Quasi-direct sums of rings and their radicals" @default.
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- W2132871052 doi "https://doi.org/10.1080/00927879708826038" @default.
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