On commutative twisted group rings

Todor Zh. Mollov; Nako A. Nachev

Czechoslovak Mathematical Journal (2005)

  • Volume: 55, Issue: 2, page 371-392
  • ISSN: 0011-4642

Abstract

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Let G be an abelian group, R a commutative ring of prime characteristic p with identity and R t G a commutative twisted group ring of G over R . Suppose p is a fixed prime, G p and S ( R t G ) are the p -components of G and of the unit group U ( R t G ) of R t G , respectively. Let R * be the multiplicative group of R and let f α ( S ) be the α -th Ulm-Kaplansky invariant of S ( R t G ) where α is any ordinal. In the paper the invariants f n ( S ) , n ∈ ℕ ∪ { 0 } , are calculated, provided G p = 1 . Further, a commutative ring R with identity of prime characteristic p is said to be multiplicatively p -perfect if ( R * ) p = R * . For these rings the invariants f α ( S ) are calculated for any ordinal α and a description, up to an isomorphism, of the maximal divisible subgroup of S ( R t G ) is given.

How to cite

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Mollov, Todor Zh., and Nachev, Nako A.. "On commutative twisted group rings." Czechoslovak Mathematical Journal 55.2 (2005): 371-392. <http://eudml.org/doc/30951>.

@article{Mollov2005,
abstract = {Let $G$ be an abelian group, $R$ a commutative ring of prime characteristic $p$ with identity and $R_tG$ a commutative twisted group ring of $G$ over $R$. Suppose $p$ is a fixed prime, $G_p$ and $S(R_tG)$ are the $p$-components of $G$ and of the unit group $U(R_tG)$ of $R_tG$, respectively. Let $R^*$ be the multiplicative group of $R$ and let $f_\alpha (S)$ be the $ \alpha $-th Ulm-Kaplansky invariant of $S(R_tG)$ where $\alpha $ is any ordinal. In the paper the invariants $f_n(S)$, $ n\in \mathbb \{N\}\cup \lbrace 0\rbrace $, are calculated, provided $G_p=1$. Further, a commutative ring $R$ with identity of prime characteristic $p$ is said to be multiplicatively $p$-perfect if $(R^*)^p = R^*$. For these rings the invariants $f_\alpha (S)$ are calculated for any ordinal $\alpha $ and a description, up to an isomorphism, of the maximal divisible subgroup of $S(R_tG)$ is given.},
author = {Mollov, Todor Zh., Nachev, Nako A.},
journal = {Czechoslovak Mathematical Journal},
keywords = {unit groups; isomorphism; Ulm-Kaplansky invariants; commutative twisted group rings; unit groups; Ulm-Kaplansky invariants; commutative twisted group rings; divisible subgroups},
language = {eng},
number = {2},
pages = {371-392},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On commutative twisted group rings},
url = {http://eudml.org/doc/30951},
volume = {55},
year = {2005},
}

TY - JOUR
AU - Mollov, Todor Zh.
AU - Nachev, Nako A.
TI - On commutative twisted group rings
JO - Czechoslovak Mathematical Journal
PY - 2005
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 55
IS - 2
SP - 371
EP - 392
AB - Let $G$ be an abelian group, $R$ a commutative ring of prime characteristic $p$ with identity and $R_tG$ a commutative twisted group ring of $G$ over $R$. Suppose $p$ is a fixed prime, $G_p$ and $S(R_tG)$ are the $p$-components of $G$ and of the unit group $U(R_tG)$ of $R_tG$, respectively. Let $R^*$ be the multiplicative group of $R$ and let $f_\alpha (S)$ be the $ \alpha $-th Ulm-Kaplansky invariant of $S(R_tG)$ where $\alpha $ is any ordinal. In the paper the invariants $f_n(S)$, $ n\in \mathbb {N}\cup \lbrace 0\rbrace $, are calculated, provided $G_p=1$. Further, a commutative ring $R$ with identity of prime characteristic $p$ is said to be multiplicatively $p$-perfect if $(R^*)^p = R^*$. For these rings the invariants $f_\alpha (S)$ are calculated for any ordinal $\alpha $ and a description, up to an isomorphism, of the maximal divisible subgroup of $S(R_tG)$ is given.
LA - eng
KW - unit groups; isomorphism; Ulm-Kaplansky invariants; commutative twisted group rings; unit groups; Ulm-Kaplansky invariants; commutative twisted group rings; divisible subgroups
UR - http://eudml.org/doc/30951
ER -

References

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