On the homogeneity of additive mappings |
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Authors: | J Rätz |
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Institution: | (1) Universität Bern, CH-3012 Bern, Schweiz |
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Abstract: | LetK be a ring with an identity 1 0 andM, L two unitaryK-modules. Then, for any additive mappingf:M L, the setH
f
:={ K f(x)=f(x) for allx M} forms a subring ofK, the homogeneity ring off. It is shown that, forM {0},L {0} and any subringS ofK for whichM is a freeS-module, there exists an additive mappingf:ML such thatH
f
=S. This result is applied to the four Cauchy functional equations, and it leads also to an answer to the question as to whether it is possible to introduce onM a multiplication ·:M × M M makingM into a ring but not into aK-algebra. |
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Keywords: | Primary 13C05 16A64 17A99 |
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