Graded Modules as a Clean Comodule

In ring and module theory, the cleanness property is well established. If any element of R can be expressed as the sum of an idempotent and a unit, then R is said to be a clean ring. Moreover, an R-module M is clean if the endomorphism ring of M is clean. We study the cleanness concept of coalgebra and comodules as a dualization of the cleanness in rings and modules. Let C be an R-coalgebra and M be a C-comodule. Since the endomorphism of C-comodule M is a ring, M is called a clean C-comodule if the ring of C-comodule endomorphisms of M is clean. In Brzezi´nski and Wisbauer (2003), the group ring R[G] is an R-coalgebra. Consider M as an R[G]-comodule. In this paper, we have investigated some sucient conditions to make M a clean R[G]-comodule, and have shown that every G-graded module M is a clean R[G]-comodule if M is a clean R-module.

Download Full-text

SOME ASPECTS OF MODULAR LATTICES WITH DUAL KRULL DIMENSION

Journal of Algebra and Its Applications ◽

10.1142/s0219498805001149 ◽

2005 ◽

Vol 04 (03) ◽

pp. 237-244

Author(s):

MARK L. TEPLY ◽

SEOG HOON RIM

Keyword(s):

Endomorphism Ring ◽

Modular Lattice ◽

Type Theorem ◽

Krull Dimension ◽

Local Domain ◽

Module Theory ◽

Modular Lattices ◽

Atomic Lattices

For an ordinal α, a modular lattice L with 0 and 1 is α-atomic if L has dual Krull dimension α but each interval [0,x] with x < 1 has dual Krull dimension <α. The properties of α-atomic lattices are presented and applied to module theory. The endomorphism ring of certain types of α-atomic modules is a local domain and hence there is a Krull–Schmidt type theorem for those α-atomic modules.

Download Full-text

Quasi-finite Irreducible Graded Modules for the Virasoro-like Algebra

Algebra Colloquium ◽

10.1142/s1005386713000175 ◽

2013 ◽

Vol 20 (02) ◽

pp. 181-196 ◽

Cited By ~ 4

Author(s):

Weiqiang Lin ◽

Yucai Su

Keyword(s):

Weight Module ◽

Highest Weight Module ◽

Graded Modules ◽

Highest Weight ◽

Finite Dimensional ◽

Irreducible Modules ◽

Graded Module ◽

Intermediate Series ◽

Uniformly Bounded

In this paper, we consider the classification of irreducible Z- and Z2-graded modules with finite-dimensional homogeneous subspaces over the Virasoro-like algebra. We prove that such a module is a uniformly bounded module or a generalized highest weight module. Then we determine all generalized highest weight quasi-finite irreducible modules. As a consequence, we determine all the modules with nonzero center. Finally, we prove that there does not exist any non-trivial Z-graded module of intermediate series.

Download Full-text

2-Clean Rings

Canadian Mathematical Bulletin ◽

10.4153/cmb-2009-017-5 ◽

2009 ◽

Vol 52 (1) ◽

pp. 145-153 ◽

Cited By ~ 2

Author(s):

Z. Wang ◽

J. L. Chen

Keyword(s):

Local Ring ◽

Group Ring ◽

Endomorphism Ring ◽

Clean Rings

AbstractA ring R is said to be n-clean if every element can be written as a sum of an idempotent and n units. The class of these rings contains clean rings and n-good rings in which each element is a sum of n units. In this paper, we show that for any ring R, the endomorphism ring of a free R-module of rank at least 2 is 2-clean and that the ring B(R) of all ω × ω row and column-finite matrices over any ring R is 2-clean. Finally, the group ring RCn is considered where R is a local ring.

Download Full-text

Nil-clean group rings

Journal of Algebra and Its Applications ◽

10.1142/s0219498817501353 ◽

2016 ◽

Vol 16 (07) ◽

pp. 1750135 ◽

Cited By ~ 4

Author(s):

Serap Sahinkaya ◽

Gaohua Tang ◽

Yiqiang Zhou

Keyword(s):

Abelian Group ◽

Nilpotent Group ◽

Group Ring ◽

Nilpotent Element ◽

General Situation ◽

Group Rings ◽

Locally Finite ◽

Arbitrary Ring ◽

Clean Ring ◽

Nil Clean Ring

An element [Formula: see text] of a ring [Formula: see text] is nil-clean, if [Formula: see text], where [Formula: see text] and [Formula: see text] is a nilpotent element, and the ring [Formula: see text] is called nil-clean if each of its elements is nil-clean. In [W. Wm. McGovern, S. Raja and A. Sharp, Commutative nil clean group rings, J. Algebra Appl. 14(6) (2015) 5; Article ID: 1550094], it was proved that, for a commutative ring [Formula: see text] and an abelian group [Formula: see text], the group ring [Formula: see text] is nil-clean, iff [Formula: see text] is nil-clean and [Formula: see text] is a [Formula: see text]-group. Here, we discuss the nil-cleanness of group rings in general situation. We prove that the group ring of a locally finite [Formula: see text]-group over a nil-clean ring is nil-clean, and that the hypercenter of the group [Formula: see text] must be a [Formula: see text]-group if a group ring of [Formula: see text] is nil-clean. Consequently, the group ring of a nilpotent group over an arbitrary ring is nil-clean, iff the ring is a nil-clean ring and the group is a [Formula: see text]-group.

Download Full-text

The group ring ℤ(p)Cq and Ye’s theorem

Journal of Algebra and Its Applications ◽

10.1142/s0219498818501116 ◽

2018 ◽

Vol 17 (06) ◽

pp. 1850111

Author(s):

Warren Wm. McGovern

Keyword(s):

Group Ring ◽

Clean Ring

We generalize Ye’s Theorem which states that the group ring [Formula: see text] is a semi-clean ring [Y. Ye, Semiclean rings, Comm. Algebra 31 (2003) 5609–5625]. The proof provided here is more efficient; it is less algorithmic but has the feature that the following statement is evident: for distinct primes [Formula: see text], the group ring [Formula: see text] is feebly clean if and only if the order of [Formula: see text] modulo [Formula: see text] is at least [Formula: see text].

Download Full-text

Some ∗-Clean Group Rings

Algebra Colloquium ◽

10.1142/s1005386715000152 ◽

2015 ◽

Vol 22 (01) ◽

pp. 169-180 ◽

Cited By ~ 6

Author(s):

Yanyan Gao ◽

Jianlong Chen ◽

Yuanlin Li

Keyword(s):

Local Ring ◽

Group Ring ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Group Rings ◽

Clean Rings ◽

Clean Ring ◽

Commutative Local Ring ◽

Necessary And Sufficient

A ring with involution ∗ is called ∗-clean if each of its elements is the sum of a unit and a projection. It is obvious that ∗-clean rings are clean. Vaš asked whether there exists a clean ring with involution that is not ∗-clean. In this paper, we investigate when a group ring RG is ∗-clean, where ∗ is the classical involution on RG. We obtain necessary and sufficient conditions for RG to be ∗-clean, where R is a commutative local ring and G is one of the groups C3, C4, S3 and Q8. As a consequence, we provide many examples of group rings which are clean but not ∗-clean.

Download Full-text

On the structure of the Group-ring of an elementary abelian group and a Theorem of Brauer and Nesbitt

Bulletin de la Classe des sciences ◽

10.3406/barb.1981.57066 ◽

1981 ◽

Vol 67 (1) ◽

pp. 260-267

Author(s):

Prabir Bhattacharya

Keyword(s):

Abelian Group ◽

Group Ring ◽

Elementary Abelian Group

Download Full-text

Families of abelian surfaces with real multiplication over Hilbert modular surfaces

Nagoya Mathematical Journal ◽

10.1017/s0027763000020080 ◽

1982 ◽

Vol 88 ◽

pp. 17-53 ◽

Cited By ~ 2

Author(s):

G. van der Geer ◽

K. Ueno

Keyword(s):

Endomorphism Ring ◽

Symplectic Group ◽

Abelian Surface ◽

Quadratic Relation ◽

Holomorphic Map ◽

Real Multiplication ◽

Compact Complex Surfaces ◽

Hilbert Modular Surfaces ◽

Hilbert Modular ◽

Abelian Functions

Around the beginning of this century G. Humbert ([9]) made a detailed study of the properties of compact complex surfaces which can be parametrized by singular abelian functions. A surface parametrized by singular abelian functions is the image under a holomorphic map of a singular abelian surface (i.e. an abelian surface whose endomorphism ring is larger than the ring of rational integers). Humbert showed that the periods of a singular abelian surface satisfy a quadratic relation with integral coefficients and he constructed an invariant D of such a relation with respect to the action of the integral symplectic group on the periods.

Download Full-text