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1.
Let R be a commutative ring with identity, M n (R) the R-algebra consisting of all n by n matrices over R. In this article, for n ≥ 5 we classify linear maps φ from M n (R) into itself satisfying φ(x)x + xφ(x) = 0 whenever x 2 = 0. We call such maps as square-zero derivations. 相似文献
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A. A. Tuganbaev 《Mathematical Notes》1993,53(2):204-207
Translated from Matematicheskie Zametki, Vol. 53, No. 2, pp. 126–131, February, 1993. 相似文献
3.
Gautam Borooah 《Journal of Pure and Applied Algebra》2008,212(1):281-296
We will completely characterize the commutative local rings for which Mn(R) is strongly clean, in terms of factorization in R[t]. We also obtain similar elementwise results which show additionally that for any monic polynomial f∈R[t], the strong cleanness of the companion matrix of f is equivalent to the strong cleanness of all matrices with characteristic polynomial f. 相似文献
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Using a growth function,GK defined for algebras over integral domains, we construct a generalization of Gelfand Kirillov dimensionGGK. GGK coincides with the classical no-tion of GK for algebras over a field, but is defined for algebras over arbitrary commutative rings. It is proved that GGK exceeds the Krull dimension for affine Noetherian PI algebras. The main result is that algebras of GGK at most one are PI for a large class of commutative Noetherian base rings including the ring of integers, Z. This extends the well-known result of Small, Stafford, and Warfield found in [11]. 相似文献
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Guillermo Cortiñas 《代数通讯》2013,41(3):1403-1412
Abstract Let D(H) be the quantum double associated to a finite dimensional quasi-Hopf algebra H, as in Hausser and Nill ((Hausser, F., Nill, F. (1999a). Diagonal crossed products by duals of quasi-quantum groups. Rev. Math. Phys. 11:553–629) and (Hausser, F., Nill, F. (1999b). Doubles of quasi-quantum groups. Comm. Math. Phys. 199:547–589)). In this note, we first generalize a result of Majid (Majid, S. (1991). Doubles of quasitriangular Hopf algebras. Comm. Algebra 19:3061–3073) for Hopf algebras, and then prove that the quantum double of a finite dimensional quasitriangular quasi-Hopf algebra is a biproduct in the sense of Bulacu and Nauwelaerts (Bulacu, D., Nauwelaerts, E. (2002). Radford's biproduct for quasi-Hopf algebras and bosonization. J. Pure Appl. Algebra 179:1–42.). 相似文献
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In the paper, we prove that if two semigroups of invertible matrices with nonnegative elements over partially ordered commutative
rings are elementarily equivalent, then their dimensions coincide and the corresponding semirings of nonnegative elements
are elementarily equivalent. 相似文献
12.
Automorphisms of upper triangular matrix rings 总被引:2,自引:0,他引:2
S. Jøndrup 《Archiv der Mathematik》1987,49(6):497-502
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Carl Faith 《Proceedings of the American Mathematical Society》1996,124(2):341-344
An overlooked corollary to the main result of the stated paper (Proc. Amer. Math. Soc. 120 (1994), 989--993) is that any Goldie ring of Goldie dimension 1 has Artinian classical quotient ring , hence is a Kerr ring in the sense that the polynomial ring satisfies the on annihilators . More generally, we show that a Goldie ring has Artinian when every zero divisor of has essential annihilator (in this case is a local ring; see Theorem ). A corollary to the proof is Theorem 2: A commutative ring has Artinian iff is a Goldie ring in which each element of the Jacobson radical of has essential annihilator. Applying a theorem of Beck we show that any ring that has Noetherian local ring for each associated prime is a Kerr ring and has Kerr polynomial ring (Theorem 5).
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Huanyin Chen 《Czechoslovak Mathematical Journal》2009,59(1):145-158
A matrix A ∈ M
n
(R) is e-clean provided there exists an idempotent E ∈ M
n
(R) such that A-E ∈ GL
n
(R) and det E = e. We get a general criterion of e-cleanness for the matrix [[a
1, a
2,..., a
n
+1]]. Under the n-stable range ondition, it is shown that [[a
1, a
2,..., a
n
+1]] is 0-clean iff (a
1, a
2,..., a
n
+1) = 1. As an application, we prove that the 0-cleanness and unit-regularity for such n × n matrix over a Dedekind domain coincide for all n ⩾ 3. The analogous for (s, 2) property is also obtained.
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Wai-Sin Ching 《Linear algebra and its applications》1977,18(3):257-266
A generalized rank (McCoy rank) of a matrix with entries in a commutative ring R with identity is discussed. Some necessary and sufficient conditions for the solvability of the linear equation Ax = b are derived, where x, b are vectors and A is a matrix with entries in either a Noetherian full quotient ring or a zero dimensional ring. 相似文献
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Young Kwon song 《代数通讯》2013,41(4):1649-1663
Maximal commutative subalgebras of the algebra of n by n matrices over a field k very rarely have dimension smaller than n. There is a (B, N)-construction which yields subalgebras of this kind. The Courter's algebra that is of this kind was shown a (B, N)-construction where B is the Schur algebra of size 4 and N = k 4. That is, the Courter's algebra is isomorphic to B ? (k 4)2, the idealization of (k 4)2. It was questioned how many isomorphism classes can be produced by varying the finitely generated faithful B-module N. In this paper, we will show that the set of all algebras B ? N 2 fall into a single isomorphism class, where B is the Schur algebra of size 4 and N a finitely generated faithful B-module. 相似文献
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E. I. Bunina 《Mathematical Notes》2012,91(1-2):3-11
In the paper, the automorphisms of the semigroup of nonnegative invertible matrices of order two over a partially ordered commutative ring with 2 invertible are described. 相似文献