On a decomposition equality in modular group rings

Danchev P. V.
Владикавказский математический журнал. . Том 9. 2007 г.. Выпуск 2.

Аннотация:
Let \(G\) be an abelian group such that \(A\leq G\) with \(p\)-component \(A_p\) and \(B\leq G\), and let \(R\) be a commutative ring with \(1\) of prime characteristic \(p\) with nil-radical \(N(R)\). It is proved that if \(A_p\not\subseteq B_p\) or \(N(R)\not= 0\), then \(S(RG)=S(RA) (1+I_p(RG; B))\) \(\iff\) \(G=AB\) and \(G_p=A_pB_p\). In particular, if \(A_p\not= 1\) or \(N(R)\not= 0\), then \(S(RG)=S(RA)\times (1+I_p(RG; B))\) \(\iff\) \(G=A\times B\). So, the question concerning the validity of this formula is completely exhausted. The main statement encompasses both the results of this type established by the author in (Hokkaido Math. J., 2000) and Miskolc Math. Notes, 2005). We also point out and eliminate in a concrete situation an error in the proof of a statement due to T. Zh. Mollov on a decomposition formula in commutative modular group rings (Proceedings of the Plovdiv University-Math., 1973).

Ключевые слова: direct factors, decompositions, normed unit groups, homomorphisms.

Язык статьи: Английский Загрузить полный текст