On Direct Decompositions in Group Near-Rings

Abstract

The following results on direct decomposition in group algebras are generalized to group near-rings.

• 1. For a family {R _i | i ∈ I} of commutative rings and an arbitrary group G:

  • (i) If $R \simeq \prod\limits_{{i \in I}} {{{R}_{i}}}$, then $RG \simeq \prod\limits_{{i \in I}} {\left( {{{R}_{i}}G} \right)}$

  • (ii) If R is a subdirect product of {R _i | i ∈ I}, then RG is a subdirect product of {R _i G | i ∈ I}.

• 2. If G is the direct product of it’s subgroups G ₁ and G ₂, then RG ≃ (RG ₁)G ₂.