A. W. Chatters, The restricted minimum condition in Noetherian hereditary rings, J. London Math. Soc. (2), 4 (1971), 83-87.
I. S. Cohen, Commutative rings with restricted minimum condition, Duke Math. J., 17 (1950), 27-42.
K. R. Goodearl and R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, London Math. Soc. Stud. Texts, 61
Cambridge University Press, Cambridge, 2004.
S. K. Jain, A. K. Srivastava and A. A. Tuganbaev, Cyclic Modules and the Structure of Rings, Oxford Math. Monogr., Oxford University Press, Oxford, 2012.
A. Karami Z. and M. R. Vedadi, On the restricted minimum condition for rings, Mediterr. J. Math., 18(1) (2021), 9 (17 pp).
A. Karami Z. and M. R. Vedadi, Restricted minimum condition for group-rings and matrix extensions, Comm. Algebra, 51(1) (2023), 168-177.
B. Kiraly, On group rings with restricted minimum condition, Ann. Math. Inform., 34 (2007), 47-49.
M. T. Koşan and J. Zemlicka, On modules and rings with the restricted minimum condition, Colloq. Math.,
140(1) (2015), 75-86.
T. Y. Lam, Lectures on Modules and Rings, Grad. Texts in Math., 189, Springer-Verlag, New York, 1999.
D. McCarthy, Infinite groups whose proper quotient groups are finite. I, Comm. Pure Appl. Math., 21 (1968), 545-562.
D. McCarthy, Infinite groups whose proper quotient groups are finite. II, Comm. Pure Appl. Math., 23 (1970), 767-789.
A. Ju. Olshanskii, Infinite groups with cyclic subgroups, (Russian), Dokl. Akad. Nauk SSSR, 245(4) (1979), 785-787.
A. Ju. Olshanskii, An infinite group with subgroups of prime orders, (Russian), Izv. Akad. Nauk. SSSR Ser. Mat., 44(2) (1980), 309-321.
A. Ju. Olshanskii, Groups of bounded period with subgroups of prime order, (Russian), Algebra i Logika, 21(5) (1982), 553-618.
A. J. Ornstein, Rings with restricted minimum condition, Proc. Amer. Math. Soc., 19 (1968), 1145-1150.
D. H. Paek, Chain conditions for subgroups of infinite order or index, J. Algebra, 249(2) (2002), 291-305.
D. J. S. Robinson, A Course in the Theory of Groups, Grad. Texts in Math., 80, Springer-Verlag, New York, 1996.
W. Rudin and H. Schneider, Idempotents in group rings, Duke Math. J., 31 (1964), 585-602.
J. S. Wilson, Groups with every proper quotient finite, Proc. Cambridge Philos. Soc., 69 (1971), 373-391.
E. I. Zel'manov, Solution of the restricted Burnside problem for groups of odd exponent, (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 54(1) (1990), 42-59, 221; translation in
Math. USSR-Izv., 36(1) (1991), 41-60.
E. I. Zel'manov, Solution of the restricted Burnside problem for 2-groups, (Russian), Mat. Sb., 182(4) (1991), 568-592; translation in
Math. USSR-Sb., 72(2) (1992), 543-565.
Restricted-finite groups with some applications in group rings
We carry out a study of groups $G$ in which the index of any infinite subgroup is finite. We call them restricted-finite groups and
characterize finitely generated not torsion restricted-finite groups. We show that every infinite restricted-finite abelian group is isomorphic to
$ \mathbb{Z}\times K$ or $\mathbb{Z}_{p^\infty}\times K$, where $K$ is a finite group and $p$ is a prime number.
We also prove that a group $G$ is infinitely generated restricted-finite
if and only if $G = AT$ where $A$ and $T$ are subgroups of $G$ such that $A$ is normal quasi-cyclic and $T$ is finite.
As an application of our results, we show that if $G$ is not torsion with finite $G'$ and the group-ring $RG$ has restricted minimum condition, then $R$ is a semisimple ring and $G\cong T\rtimes\mathbb{Z} $, where $T$ is finite whose order is unit in $R$.
The converse is also true with certain conditions including $G = T\times \mathbb{Z} $.
A. W. Chatters, The restricted minimum condition in Noetherian hereditary rings, J. London Math. Soc. (2), 4 (1971), 83-87.
I. S. Cohen, Commutative rings with restricted minimum condition, Duke Math. J., 17 (1950), 27-42.
K. R. Goodearl and R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, London Math. Soc. Stud. Texts, 61
Cambridge University Press, Cambridge, 2004.
S. K. Jain, A. K. Srivastava and A. A. Tuganbaev, Cyclic Modules and the Structure of Rings, Oxford Math. Monogr., Oxford University Press, Oxford, 2012.
A. Karami Z. and M. R. Vedadi, On the restricted minimum condition for rings, Mediterr. J. Math., 18(1) (2021), 9 (17 pp).
A. Karami Z. and M. R. Vedadi, Restricted minimum condition for group-rings and matrix extensions, Comm. Algebra, 51(1) (2023), 168-177.
B. Kiraly, On group rings with restricted minimum condition, Ann. Math. Inform., 34 (2007), 47-49.
M. T. Koşan and J. Zemlicka, On modules and rings with the restricted minimum condition, Colloq. Math.,
140(1) (2015), 75-86.
T. Y. Lam, Lectures on Modules and Rings, Grad. Texts in Math., 189, Springer-Verlag, New York, 1999.
D. McCarthy, Infinite groups whose proper quotient groups are finite. I, Comm. Pure Appl. Math., 21 (1968), 545-562.
D. McCarthy, Infinite groups whose proper quotient groups are finite. II, Comm. Pure Appl. Math., 23 (1970), 767-789.
A. Ju. Olshanskii, Infinite groups with cyclic subgroups, (Russian), Dokl. Akad. Nauk SSSR, 245(4) (1979), 785-787.
A. Ju. Olshanskii, An infinite group with subgroups of prime orders, (Russian), Izv. Akad. Nauk. SSSR Ser. Mat., 44(2) (1980), 309-321.
A. Ju. Olshanskii, Groups of bounded period with subgroups of prime order, (Russian), Algebra i Logika, 21(5) (1982), 553-618.
A. J. Ornstein, Rings with restricted minimum condition, Proc. Amer. Math. Soc., 19 (1968), 1145-1150.
D. H. Paek, Chain conditions for subgroups of infinite order or index, J. Algebra, 249(2) (2002), 291-305.
D. J. S. Robinson, A Course in the Theory of Groups, Grad. Texts in Math., 80, Springer-Verlag, New York, 1996.
W. Rudin and H. Schneider, Idempotents in group rings, Duke Math. J., 31 (1964), 585-602.
J. S. Wilson, Groups with every proper quotient finite, Proc. Cambridge Philos. Soc., 69 (1971), 373-391.
E. I. Zel'manov, Solution of the restricted Burnside problem for groups of odd exponent, (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 54(1) (1990), 42-59, 221; translation in
Math. USSR-Izv., 36(1) (1991), 41-60.
E. I. Zel'manov, Solution of the restricted Burnside problem for 2-groups, (Russian), Mat. Sb., 182(4) (1991), 568-592; translation in
Math. USSR-Sb., 72(2) (1992), 543-565.
Taerı, B., & Vedadı, M. R. (2024). Restricted-finite groups with some applications in group rings. International Electronic Journal of Algebra, 36(36), 51-65. https://doi.org/10.24330/ieja.1438622
AMA
Taerı B, Vedadı MR. Restricted-finite groups with some applications in group rings. IEJA. July 2024;36(36):51-65. doi:10.24330/ieja.1438622
Chicago
Taerı, Bijan, and Mohammad Reza Vedadı. “Restricted-Finite Groups With Some Applications in Group Rings”. International Electronic Journal of Algebra 36, no. 36 (July 2024): 51-65. https://doi.org/10.24330/ieja.1438622.
EndNote
Taerı B, Vedadı MR (July 1, 2024) Restricted-finite groups with some applications in group rings. International Electronic Journal of Algebra 36 36 51–65.
IEEE
B. Taerı and M. R. Vedadı, “Restricted-finite groups with some applications in group rings”, IEJA, vol. 36, no. 36, pp. 51–65, 2024, doi: 10.24330/ieja.1438622.
ISNAD
Taerı, Bijan - Vedadı, Mohammad Reza. “Restricted-Finite Groups With Some Applications in Group Rings”. International Electronic Journal of Algebra 36/36 (July 2024), 51-65. https://doi.org/10.24330/ieja.1438622.
JAMA
Taerı B, Vedadı MR. Restricted-finite groups with some applications in group rings. IEJA. 2024;36:51–65.
MLA
Taerı, Bijan and Mohammad Reza Vedadı. “Restricted-Finite Groups With Some Applications in Group Rings”. International Electronic Journal of Algebra, vol. 36, no. 36, 2024, pp. 51-65, doi:10.24330/ieja.1438622.
Vancouver
Taerı B, Vedadı MR. Restricted-finite groups with some applications in group rings. IEJA. 2024;36(36):51-65.