Abstract

Automorphisms and derivations of associative rings by V. Kharchenko

By V. Kharchenko

T moi, ... si favait su remark en revenir. One sel'Yice arithmetic has rendered the je n'y serais element aile.' human race. It has placed logic again Jules Verne the place it belongs, at the topmost shelf subsequent to the dusty canister labelled 'discarded non- The sequence is divergent; accordingly we will be sense', in a position to do whatever with it. Eric T. Bell O. Heaviside arithmetic is a device for proposal. A hugely helpful instrument in an international the place either suggestions and non linearities abound. equally, every kind of components of arithmetic function instruments for different elements and for different sciences. using an easy rewriting rule to the quote at the correct above one unearths such statements as: 'One provider topology has rendered mathematical physics .. .'; 'One provider good judgment has rendered com puter technology .. .'; 'One provider class concept has rendered arithmetic .. .'. All arguably real. And all statements accessible this manner shape a part of the raison d 'e\re of this sequence.

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Extra resources for Automorphisms and derivations of associative rings

Example text

1. Proposition. L be a derivation determined on the essential ideal I of a semiprime ring R, with its values in R. L can be uniquely extended to the derivation of ring Q and that of ring RF · Proof. :: R Let r E R F' and denote the ideal ( I I rl homomorphism of the left ~( ]) = ( Let us consider an ideal I-modules through ~: J J. IrE IF, such that Let us consider the -+ R, acting by the formula jr) JL - j JL r. It should be remarked that Jr ~ (jr)JL is determined. JL rE R. 9), we find an element jrI = (jr)JL - jJL r.

This idempotent will here from be denoted by e( m) and called a support of element m, which fact does not contradict the denotation 8(S) for elements S E R F assumed above. 22. Lemma. Let M be a complete nonsingular c-module, N be a closed submodule of M. Then factor-module M / N is complete nonsingular module. Proof. 16 the module N is injective and, hence, it is singled out from M by a direct addent: M = N $ K. One, has, finally, to remark that M / N == K is an injective nonsingular module. 23.

The approach for presenting such a classical material is clear to be quite different. We prefer a topological approach by an important reason that all module homomorphisms, ring automorphisms and derivations, as well as termal unary operations prove to be continuous and, hence, the problem of transferring many properties onto an injective hull (or, in our terms, onto a topological closure) proves to be absolutely evident. In the case of a prime ring all the modules considered here are converted into linear spaces, and the topology becomes discrete.

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