By Catarina Santa-Clara

Surveying the main influential advancements within the box, this complaints studies the newest examine on algebras and their representations, commutative and non-commutative earrings, modules, conformal algebras, and torsion theories. the quantity collects stimulating discussions from world-renowned names together with Tsit-Yuen Lam, Larry Levy, Barbara Osofsky, and Patrick Smith.

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**Additional resources for Algebras, Rings And Their Representations: Proceedings Of The International Conference on Algebras, Modules and Rings, Lisbon, Portugal, 14-18 July 2003**

**Example text**

Primeness of T gives either h = 0 or Ji = 0. 7 we conclude I0 = 0 and therefore 1 = 0. In the second case we similarly obtain J = 0. Hence L is gr-prime. Conversely, let us now suppose L is gr-prime and let I, J be two ideals of T satisfying [I, T, J] + [J, T, I] = 0. It is easy to check that [I, T]®I and [J,T}®J are gr-ideals of L. We assert [[/,T]@I, [J,T)®J} = 0. Indeed, for any x S [/,T] ® I we have x = ( X S i £(xi'x'i)>z) w ^ n z>Xi £ I, x't £T for i = 1 , . . , ny. ly3>ypx'i\>xi)-c([yj>y'j>xi}>x'i))- E (2) »ii=l We have, for any u e T , i e { 1 , .

6]) and unknown for Lie algebras. Thus, the study of the inheritance of regularity by ideals in Jordan and Lie algebras requires the use of different techniques. In 1984 McCrimmon began the study of ideals in strongly prime Jordan systems (see [4]). Subsequently, he improved this work in [5] where the hypothesis of nondegenerancy was removed and only semiprimeness of the ideals was required for the inheritance of primeness. To do so, he introduced the notion of the eventual annihilator of an ideal I as the union of the annihilators of the powers of / .

Kn G K such that K C Wko U • • • U Wkn. Then, evidently, i f - L / C W, where(7 = [/ ( f c o ) n---nC/ ( A ; " ) . • Lemma 7. Let Rbe a topological ring and h(R) < m, where m is an infinite cardinal number. Let (M, T) G TopMod and V be a ^^-neighborhood, V = Ua£nKa, |fi| < m, where each Ka is a compact subset. Then for every 0-neighborhood W of (M, 7^) there exists a 0-neighborhood U of (M, 7^,) such that V-U CW. Proof. We can assume without loss of generality that W = C\aGQWa, where each Wa is a O-neighborhood of (M,T).