Geodesic
On a decomposition equality in modular group rings
Vladikavkazskij matematičeskij žurnal, Tome 9 (2007) no. 2, pp. 3-8 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Let $G$ be an abelian group such that $A\le G$ with $p$-component $A_p$ and $B\le G$, 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 $N(R)\ne 0$, then $S(RG)=S(RA)(1+I_p(RG;B))$ $\iff$ $G=AB$ and $G_p=A_pB_p$. In particular, if $A_p\ne 1$ or $N(R)\ne 0$, 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). 
@article{VMJ_2007_9_2_a0,
     author = {P. V. Danchev},
     title = {On a decomposition equality in modular group rings},
     journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
     pages = {3--8},
     year = {2007},
     volume = {9},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/VMJ_2007_9_2_a0/}
}
TY  - JOUR
AU  - P. V. Danchev
TI  - On a decomposition equality in modular group rings
JO  - Vladikavkazskij matematičeskij žurnal
PY  - 2007
SP  - 3
EP  - 8
VL  - 9
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/VMJ_2007_9_2_a0/
LA  - en
ID  - VMJ_2007_9_2_a0
ER  - 
%0 Journal Article
%A P. V. Danchev
%T On a decomposition equality in modular group rings
%J Vladikavkazskij matematičeskij žurnal
%D 2007
%P 3-8
%V 9
%N 2
%U http://geodesic.mathdoc.fr/item/VMJ_2007_9_2_a0/
%G en
%F VMJ_2007_9_2_a0
P. V. Danchev. On a decomposition equality in modular group rings. Vladikavkazskij matematičeskij žurnal, Tome 9 (2007) no. 2, pp. 3-8. http://geodesic.mathdoc.fr/item/VMJ_2007_9_2_a0/

[1] Danchev P., “Isomorphism of modular group algebras of direct sums of torsion-complete abelian $p$-groups”, Rend. Sem. Mat. Univ. Padova, 101 (1999), 51–58 | MR | Zbl

[2] Danchev P., “The splitting problem and the direct factor problem in modular abelian group algebras”, Math. Balkanica, 14:3/4 (2000), 217–226 | MR | Zbl

[3] Danchev P., “Modular group algebras of coproducts of countable abelian groups”, Hokkaido Math. J., 29:2 (2000), 255–262 | MR | Zbl

[4] Danchev P., “Commutative group algebras of cardinality $\aleph_1$”, Southeast Asian Bull. Math., 25:4 (2001/2002), 589–598 | DOI | MR

[5] Danchev P., “The direct factor problem for modular group algebras of isolated direct sums of torsion-complete abelian $p$-groups”, Southeast Asian Bull. Math., 26:4 (2002), 559–565 | DOI | MR | Zbl

[6] Danchev P., “Commutative group algebras of direct sums of $\sigma$-summable abelian $p$-groups”, Math. J. Okayama Univ., 45:2 (2003), 1–15 | MR | Zbl

[7] Danchev P., “Commutative group algebras of direct sums of countable abelian groups”, Kyungpook Math. J., 44:1 (2004), 21–29 | MR | Zbl

[8] Danchev P., “Algebraically compactness of Sylow $p$-groups in abelian group rings of characteristic $p$”, Riv. Mat. Univ. Parma, 3 (2004), 61–67 | MR | Zbl

[9] Danchev P., “On a decomposition formula in commutative group rings”, Miskolc Math. Notes, 6:2 (2005), 153–159 | MR | Zbl

[10] Danchev P., “Basic subgroups in modular abelian group algebras”, Czechoslovak Math. J., 57:1 (2007), 173–182 | DOI | MR | Zbl

[11] Mollov T. Zh., “Pure subgroups and a separation of direct factors in the unit group of modular group algebras of abelian $p$-groups”, N. tr. Plovdiv Univ.-Math., 11:1 (1973), 9–15 (In Bulgarian) | MR

[12] Mollov T. Zh., Zbl. Math., 959.20003

[13] Mollov T. Zh., Nachev N. A., “Unit groups of commutative group rings”, Comm. Alg., 34:10 (2006), 3835–3857 | DOI | MR | Zbl