By J. L. S. Chen, N. Q. Ding
The chosen papers during this quantity conceal all of the most vital parts of ring thought and module conception akin to classical ring concept, illustration conception, the speculation of quantum teams, the speculation of Hopf algebras, the idea of Lie algebras and Abelian workforce idea. The assessment articles, written via experts, offer a great assessment of many of the parts of ring and module idea - perfect for researchers trying to find a brand new or comparable box of research. additionally incorporated are unique articles exhibiting the rage of present examine.
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Additional resources for Advances in Ring Theory - Proceedings of: Proceedings of the 4th China-Japan-Korea International Conference, Nanjing, China 24-28 June 2004
By Erlich , an element a 6 R is unit 26 27 regular if and only if a is both (von Neumann) regular and left morphic. n(R) is left morphic. Right morphic rings and strongly right morphic rings are defined analogously. A left and right morphic ring is called a morphic ring. A strongly morphic ring means a strongly left and strongly right morphic ring. Left morphic rings were first introduced by Nicholson and Sanchez Campos  and were discussed in great detail in ,  and . The goal of this paper is to construct new examples of morphic rings.
Thus we say that E(G) is the endomorphism near-ring of the group G. For the remainder basic concepts and results on ring and near-ring case, we refer to ,  and . 2. R-groups. R-group, and then investigate their properties. For any ring R, right E-modules M and N, the set of all /^-module homomorphisms from M to N is denoted by HorriR(M,N) and the set of all group homomorphisms from M to N is Hom(M,N):=Homz(M,N), in particular we denote that Endn(M} := HortiR(M,M) and End(M) := Endz(M), In this case, M is called a GM module over R if every group homomorphism of M is an R-module homomorphism, that is, End(M) = EndR(M).
Thus the 'if part' is proved. Now we will prove 'only if part'. Assume that G is an MR group and assume to the contrary that there is a in G and r in R such that ar ^ a. Define a mapping / : G —> G given by (ar)/ = a and (x)f = x for all x £ G which is not equal to or. Then clearly, / € M(G), however, (ar)/ = a ^ ar = (a)/r. This implies that / is not an ^-homogeneous map, a contradiction. 14. 4, (b) in ). We call these near-rings left thread near-rings. Every regular R-group which is left thread is an MRgroup.