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Search: MSC category 16U60 ( Units, groups of units )

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1. CMB Online first

Koşan, Tamer; Sahinkaya, Serap; Zhou, Yiqiang
Additive maps on units of rings
Let $R$ be a ring. A map $f: R\rightarrow R$ is additive if $f(a+b)=f(a)+f(b)$ for all elements $a$ and $b$ of $R$. Here a map $f: R\rightarrow R$ is called unit-additive if $f(u+v)=f(u)+f(v)$ for all units $u$ and $v$ of $R$. Motivated by a recent result of Xu, Pei and Yi showing that, for any field $F$, every unit-additive map of ${\mathbb M}_n(F)$ is additive for all $n\ge 2$, this paper is about the question when every unit-additive map of a ring is additive. It is proved that every unit-additive map of a semilocal ring $R$ is additive if and only if either $R$ has no homomorphic image isomorphic to $\mathbb Z_2$ or $R/J(R)\cong \mathbb Z_2$ with $2=0$ in $R$. Consequently, for any semilocal ring $R$, every unit-additive map of ${\mathbb M}_n(R)$ is additive for all $n\ge 2$. These results are further extended to rings $R$ such that $R/J(R)$ is a direct product of exchange rings with primitive factors Artinian. A unit-additive map $f$ of a ring $R$ is called unit-homomorphic if $f(uv)=f(u)f(v)$ for all units $u,v$ of $R$. As an application, the question of when every unit-homomorphic map of a ring is an endomorphism is addressed.

Keywords:additive map, unit, 2-sum property, semilocal ring, exchange ring with primitive factors Artinian
Categories:15A99, 16U60, 16L30

2. CMB 2016 (vol 59 pp. 652)

Su, Huadong
On the Diameter of Unitary Cayley Graphs of Rings
The unitary Cayley graph of a ring $R$, denoted $\Gamma(R)$, is the simple graph defined on all elements of $R$, and where two vertices $x$ and $y$ are adjacent if and only if $x-y$ is a unit in $R$. The largest distance between all pairs of vertices of a graph $G$ is called the diameter of $G$, and is denoted by ${\rm diam}(G)$. It is proved that for each integer $n\geq1$, there exists a ring $R$ such that ${\rm diam}(\Gamma(R))=n$. We also show that ${\rm diam}(\Gamma(R))\in \{1,2,3,\infty\}$ for a ring $R$ with $R/J(R)$ self-injective and classify all those rings with ${\rm diam}(\Gamma(R))=1$, 2, 3 and $\infty$, respectively.

Keywords:unitary Cayley graph, diameter, $k$-good, unit sum number, self-injective ring
Categories:05C25, 16U60, 05C12

3. CMB 2016 (vol 59 pp. 661)

Ying, Zhiling; Koşan, Tamer; Zhou, Yiqiang
Rings in Which Every Element is a Sum of Two Tripotents
Let $R$ be a ring. The following results are proved: $(1)$ every element of $R$ is a sum of an idempotent and a tripotent that commute iff $R$ has the identity $x^6=x^4$ iff $R\cong R_1\times R_2$, where $R_1/J(R_1)$ is Boolean with $U(R_1)$ a group of exponent $2$ and $R_2$ is zero or a subdirect product of $\mathbb Z_3$'s; $(2)$ every element of $R$ is either a sum or a difference of two commuting idempotents iff $R\cong R_1\times R_2$, where $R_1/J(R_1)$ is Boolean with $J(R_1)=0$ or $J(R_1)=\{0,2\}$, and $R_2$ is zero or a subdirect product of $\mathbb Z_3$'s; $(3)$ every element of $R$ is a sum of two commuting tripotents iff $R\cong R_1\times R_2\times R_3$, where $R_1/J(R_1)$ is Boolean with $U(R_1)$ a group of exponent $2$, $R_2$ is zero or a subdirect product of $\mathbb Z_3$'s, and $R_3$ is zero or a subdirect product of $\mathbb Z_5$'s.

Keywords:idempotent, tripotent, Boolean ring, polynomial identity $x^3=x$, polynomial identity $x^6=x^4$, polynomial identity $x^8=x^4$
Categories:16S50, 16U60, 16U90

4. CMB 2010 (vol 54 pp. 237)

Creedon, Leo; Gildea, Joe
The Structure of the Unit Group of the Group Algebra ${\mathbb{F}}_{2^k}D_{8}$
Let $RG$ denote the group ring of the group $G$ over the ring $R$. Using an isomorphism between $RG$ and a certain ring of $n \times n$ matrices in conjunction with other techniques, the structure of the unit group of the group algebra of the dihedral group of order $8$ over any finite field of chracteristic $2$ is determined in terms of split extensions of cyclic groups.

Categories:16U60, 16S34, 20C05, 15A33

5. CMB 2005 (vol 48 pp. 80)

Herman, Allen; Li, Yuanlin; Parmenter, M. M.
Trivial Units for Group Rings with $G$-adapted Coefficient Rings
For each finite group $G$ for which the integral group ring $\mathbb{Z}G$ has only trivial units, we give ring-theoretic conditions for a commutative ring $R$ under which the group ring $RG$ has nontrivial units. Several examples of rings satisfying the conditions and rings not satisfying the conditions are given. In addition, we extend a well-known result for fields by showing that if $R$ is a ring of finite characteristic and $RG$ has only trivial units, then $G$ has order at most 3.

Categories:16S34, 16U60, 20C05

6. CMB 2001 (vol 44 pp. 27)

Goodaire, Edgar G.; Milies, César Polcino
Normal Subloops in the Integral Loop Ring of an $\RA$ Loop
We show that an $\RA$ loop has a torsion-free normal complement in the loop of normalized units of its integral loop ring. We also investigate whether an $\RA$ loop can be normal in its unit loop. Over fields, this can never happen.

Categories:20N05, 17D05, 16S34, 16U60

7. CMB 2000 (vol 43 pp. 60)

Farkas, Daniel R.; Linnell, Peter A.
Trivial Units in Group Rings
Let $G$ be an arbitrary group and let $U$ be a subgroup of the normalized units in $\mathbb{Z}G$. We show that if $U$ contains $G$ as a subgroup of finite index, then $U = G$. This result can be used to give an alternative proof of a recent result of Marciniak and Sehgal on units in the integral group ring of a crystallographic group.

Keywords:units, trace, finite conjugate subgroup
Categories:16S34, 16U60

8. CMB 1997 (vol 40 pp. 103)

Riley, David M.; Tasić, Vladimir
The transfer of a commutator law from a nil-ring to its adjoint group
For every field $F$ of characteristic $p\geq 0$, we construct an example of a finite dimensional nilpotent $F$-algebra $R$ whose adjoint group $A(R)$ is not centre-by-metabelian, in spite of the fact that $R$ is Lie centre-by-metabelian and satisfies the identities $x^{2p}=0$ when $p>2$ and $x^8=0$ when $p=2$. The existence of such algebras answers a question raised by A.~E.~Zalesskii, and is in contrast to positive results obtained by Krasilnikov, Sharma and Srivastava for Lie metabelian rings and by Smirnov for the class Lie centre-by-metabelian nil-algebras of exponent 4 over a field of characteristic 2 of cardinality at least 4.

Categories:16U60, 17B60

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