Advances in Ring Theory by J. L. S. Chen, N. Q. Ding

By J. L. S. Chen, N. Q. Ding

The chosen papers during this quantity conceal the entire most crucial components of ring conception and module concept reminiscent of classical ring idea, illustration thought, the speculation of quantum teams, the idea of Hopf algebras, the speculation of Lie algebras and Abelian crew conception. The overview articles, written by means of experts, offer a superb evaluation of many of the components of ring and module idea - excellent for researchers trying to find a brand new or similar box of analysis. additionally integrated are unique articles exhibiting the rage of present learn.

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Let R be any near-ring and G an additive group. Then G is called an R-group if there exists a near-ring homomorphism Such a homomorphism 6 is called a representation of R on G, we write xr for x(0r) for all x 6 G and r e R. R-group G is called unitary. R-group called the regular group. Naturally, every group G has an M(G)-group structure by applying the / 6 M(G) to the x £ G as a scalar multiplication x f . R-group G with (G, +) is abelian is called an abelian R-group. For example, if (G, +) is abelian, then M(G) is an abelian near-ring and moreover, G is an abelian M(G)-group.

Let {G, | i € A} be any family of R-groups. Then each Gi is an MR group if and only if G :— © Gj is an MR group. 11. 16. Let G be an R-group. (1) // G is a faithful MR group, then R is a commutative near-ring. (2) // G is a faithful distributive abelian MR group, then R is a commutative ring. Proof. Let a, 6 € R. Define a mapping / : G —> G given by xf = xa, for all x e G. Then clearly, / € M(G). Since G is an MR group, / € C(R;G). Thus we have the equalities: (xb)f = (xb)a = x(ba) and since / € M(G) = C(R; G), (xb)f = (xf)b = (xa)b = x(ab).

S. K. Jain, P. Kanwar, S. Malik and J. B. Srivastava, KDtx is a CS-algebra, Proc. Amer. Math. Soc. 128 (2000), 397-400. 26. I. Kaplansky, Rings of Operators, Benjamin, New York, 1968. 27. T. Y. Lam, Lectures on Modules and Rings, Springer-Verlag, Berlin- 25 Heidelberg-New York, 1999. 28. J. Lawrence, A singular primitive ring, Proc. Amer. Math. Soc. 45 (1974), 59-62. 29. A. C. Mewborn, Regular rings and Baer rings, Math. Z. 121 (1971), 211-219. 30. J. Okninski, Semigroup Algebras, Marcel Dekker, New York, 1991.

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