Quasi-direct sums of rings and their radicals |
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Authors: | YQ Chen KP Shum |
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Institution: | Department of Mathematics , The Chinese University of Hong kong , Hong Kong |
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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. |
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