Let R, S be RINGs with identity and M be a unitary (R, S)-bimodule. We characterize homomorphisms and derivations of the generalized matrix RING T =(or sm), and provide a triangular representation of the differential POLYNOMIAL RING T[b; d].

Let R be a RING with an endomorphism  and an -derivation . Antoine studied the structure of the set of nilpotent elements in Armendariz RINGs and introduced nil-Armendariz RINGs. In this paper we introduce and investigate the notion of nil-( ,  )-compatible RINGs. The class of nil-( ,  )-compatible RINGs are extended through various RING extensions and many classes of nil-( ,  )-compatible RINGs are constructed. We also prove that, if R is nil--compatible and nil-Armendariz RING of power series type with nil  R  nilpotent, then nil(R[[x;  ]])  nil(R)[[x;  ]]. We show that, if R is a nil-Armendariz RING of power series type, with nil  R  nilpotent and nil-( ,  )-compatible RING, then nil  R  x;  ,     nil  R   x;  ,   . As a consequence, several known results are unified and extended to the more general setting. Also examples are provided to illustrate our results.

Spectral synthesis deals with the description of translation invariant function spaces. It turns out that the basic building blocks of this description are the exponential monomials, which are built up from exponential functions and POLYNOMIAL functions. The author collaborated with Laczkovich [Math. Proc. Cambridge Philos. Soc. 143 (2007), no. 1, 103–120] proved that spectral synthesis holds on an Abelian group if and only if the torsion free rank of the group is finite. The author [Aequationes Math. 70 (2005), no. 1-2, 122–130] showed that the torsion free rank of an Abelian group is strongly related to the properties of POLYNOMIAL functions on the group. Here we show that spectral synthesis holds on an Abelian group if and only if the RING of POLYNOMIAL functions on the group is Noetherian.

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