International Conference on Rings and Algebras
In honor of Professor P.-H. Lee
July 10–16, 2011 (National Taiwan University, Taipei, Taiwan)

On a possibly new construction in algebraic K-Theory of rings (Talk slides)

Abstract. In a classical algebraic K-Theory of rings the definitions of the Grothendieck group and the Whitehead group are given first in case of unital rings. In case of nonunital rings, the definitions use unitisation and projections. In a paper in 2008 we offered a new approach by defining these groups a bit differently but so that the definition for unital and nonunital rings would be direct and the same fromula would hold for both cases. We were able to show that this modified construction coincides with the classical one for unital rings. For nonunital rings the question is still open. We will present the modified version and compare it with the classical one.

A weaker form of pseudo-injective modules (Talk slides)

Abstract. For a given right $R$-module $M,$ a right $R$-module $N$ is called a pseudo-$M$-c-injective module if every monomorphism from a closed submodule of $M$ to $N$ can be extended to a homomorphism from $M$ to $N$. A right $R$-module $M$ is said to be a quasi-pseudo-c-injective module if it is pseudo-$M$-c-injective. We give several properties of pseudo-$M$-c-injective and quasi-pseudo-c-injective. Beside survey several properties of pseudo-$M$-c-injective, we find out some relations of quasi-pseudo-c-injectivity with some notation such as CS-modules, uniform modules.

Decompositions of quotient rings and $m$-commuting maps (Talk slides)

Abstract. Let $R$ be a semiprime ring with symmetric Martindale quotient ring $Q$, $n\geq2$ and let $f(X)=X^nh(X)$, where $h(X)$ is a polynomial over the ring of integers with $h(0)=\pm 1$. Then there is a ring decomposition $Q=Q_1\oplus Q_2\oplus Q_3$ such that $Q_1$ is a ring satisfying $S_{2n-2}$, the standard identity of degree $2n-2$, $Q_2\cong M_n(E)$ for some commutative regular self-injective ring $E$ such that, for some fixed $q>1$, $x^q=x$ for all $x\in E$, and $Q_3$ is a both faithful $S_{2n-2}$-free and faithful $f$-free ring. Applying the theorem, we characterize $m$-commuting maps, which are defined by linear generalized differential polynomials, on a semiprime ring.

Kernel inclusions of algebraic automorphisms (Talk slides)

Abstract. Let $R$ be a prime ring with extended centroid $C$ and symmetric Martindale quotient ring $Q$. Define, for an automorphism $\sigma$ of $R$, $R^{(\sigma)}=\{x\in R\mid x^\sigma=x\}$. Let $\sigma$ and $\tau$ be automorphisms of $R$ and assume that $\sigma$ is $C$-algebraic. We show that $R^{(\sigma)} \subseteq R^{(\tau)}$ if and only if $x^{\tau}=v x^{\sigma^i} v^{-1}$ for all $x \in R$, where $i$ is an integer and where $v$ is in the centralizer of $R^{(\sigma)}$ in $Q$.

Bounded distance preserving surjective mappings on block triangular matrix algebras

Abstract. Let $M_{n}$ be the algebra of $n\times n$ square matrices. Let $T$ and $U$ be block triangular matrix subalgebras of $M_{n}$ and $M_{m}$, respectively. Let $r$ be an integer such that $1\leq r<\text{min}(\lfloor\frac{n+1}{2}\rfloor,\lfloor\frac{m+1}{2}\rfloor)$. In this talk, we show that every surjective mappings $\psi:T\to U$ satisfying $$\text{rank}(A-B)\leq r\Leftrightarrow\text{rank}(\psi(A)-\psi(B))\leq r$$ are bijective mappings preserving adjacency in both directions.

On recursion operators for elliptic models (Talk slides)

Abstract. New quasilocal recursion and Hamiltonian operators for the Krichever–Novikov and the Landau–Lifshitz equations are found. It is shown that the associative algebra of quasilocal recursion operators for these models is generated by a couple of operators related by an elliptic curve equation. A theoretical explanation of this fact for the Landau–Lifshitz equation is given in terms of multiplicators of the corresponding Lax structure.

Partial actions of groups on semiprime rings (Talk slides)

Abstract. Partial actions of groups have been studied and applied first in $C^{*}$-algebras and then in several other areas of mathematics. In a pure algebraic context, partial actions of groups on algebras have been introduced and studied by M. Dokuchaev and R. Exel¹.
 
In this survey lecture we recall the definition of partial actions. We consider, in particular, partial actions of groups on semiprime rings and study conditions under which a partial action in this case has an enveloping action (see ^{2, 3}).

Additive set of idempotents in rings (Talk slides)

Abstract. Let $R$ be a ring with identity $1$, $I(R)$ be the set of all onunits idempotents in $R$ and $M(R)$ be the set of all primitive idempotents and $0$ of $R$. $I(R)$ is said to be additive if for all $e, f \in I(R)$ $(e \neq f)$, $e + f \in I(R)$ and $M(R)$ is said to be additive in $I(R)$ if for all $e, f \in M(R)$ $(e\neq f)$, $e + f \in I(R)$. In this paoer, the following are shown: (1) $I(R)$ is additive if and only if $I(R)$ is multiplicative and the characteristic of $R$ is $2$; $M(R)$ is additive in $I(R)$ if and only if $M(R)$ is orthogonal; If $0 \neq ef \in I(R)$ for some $e \in M(R)$ and $f \in I(R)$, then $ef \in M(R)$; (2) If $R$ has a finite complete set of orthogonal primitive idempotents, then every nonzero idempotent is a sum of finite number of orthogonal primitive idempotents if and only $I(R)$ is multiplicative if and only if $M(R)$ is additive in $I(R)$.

On formal power series over Rickart rings (Talk slides)

Abstract. A ring $R$ is called a (right) Rickart ring if the right annihilator of any element in $R$ is generated, as a right ideal, by an idempotent. This definition is equivalent to that every principle right ideal is projective, and thus a (right) Rickart ring is also known as a (right) PP ring.
 
Armendariz and Jondrup had shown that if $R$ is a reduced or commutative ring, then the polynomial ring $(R[x],+,\cdot)$ is a PP ring if and only if R is a PP ring. However, this result is not true if the polynomial ring is replaced by the formal power series ring $(R[[x]], +, \cdot)$.
 
Birkenmeier, Kim and Park had introduced (right) principally quasi-Baer rings as a generalized for Rickart rings. A ring $R$ is callled (right) p.q.-Baer if the right annihilator of a principal right ideal is generated by an idempotent. They also shown that: A ring $R$ is right p.q.-Baer if and only if the polynomial ring $(R[x], +, \cdot)$ is right p.q.-Baer. Again, this result is not true if the polynomial ring is replaced by the formal power series ring $(R[[x]], +, \cdot)$. We call the right p.-q-Baer rings as right quasi-Rickart rings.
 
In this note, we discuss the conditions to guarantee the Rickart or quasi-Rickart condition be extended to the formal power series ring $(R[[x]], +, \cdot)$ or the nearring $(R_0[[x]], +, \circ)$.

Almost Injective Modules (Talk slides)

Abstract. We will give a selected survey of theory of almost injective modules introduced by Harada, endomorphism rings of indecomposable almost injective modules, direct sum of indecomposable almost injective modules, almost quasi-Frobenius rings and almost self-injective group algebras. Much of the contribution in the theory of almost injective and its dual notion almost projective is due to Harada, Oshiro and their co-workers.

Preserver problems in quantum information science (Talk slides)

Abstract. We describe some recent results and questions on preserver problems related to quantum information science.

Completely bounded disjointness preserving operators between Fourier algebras and their cb-extensions (Talk slides)

Abstract. The notion of Fourier and Fourier-Stieltjes algebras of locally compact groups were given by Eymard in 1964. They are both preduals of von Neumann algebras, and as such they possess natural operator space structures. In this talk, I will briefly introduce the Fourier and the Fourier-Stieltjes algebra first and then I will characterize surjective completely bounded disjointness preserving linear maps on the Fourier algebra of locally compact amenable groups. Moreover, we show that such a linear operator has a canonical cb-extension on the Fourier-Stielejes algebra.

Lie rings of (anti-)symmetric derivations of commutative rings with involution

Abstract. We investigate the simplicity, primeness and semiprimeness of the Lie rings of symmetric derivations of commutative rings with involution. The analogous results for (anti-)symmetric derivations are also discussed.

Certain additive maps on $m$-power closed Lie ideals (Talk slides)

Abstract. Let $R$ be a prime ring with extended centroid $C$ and $m$ a fixed positive integer $>1$. A Lie ideal $L$ of $R$ is called $m$-power closed if $u^m\in L$ for all $u\in L$. We prove that if $\text{char} R=0$ or a prime $p>m$, then every non-central, $m$-power closed Lie ideal $L$ of $R$ contains a nonzero ideal of $R$ except when $\dim_CRC=4$, $m$ is odd, and $u^{m-1}\in C$ for all $u\in L$. Moreover, the additive maps $d\colon L\to R$ satisfying $d(u^m)=mu^{m-1}d(u)$ (resp. $d(u^m)=u^{m-1}d(u)$) for all $u\in L$ are completely characterized if $\text{char} R=0$ or a prime $p>2(m-1)$.

On Bavula conjecture (Talk slides)

Abstract. In the zero characteristic case the Weyl algebra $A_{n}$ is simple and because of that any homomorphism of this algebra is an embedding. Vladimir Bavula conjectured that though $A_{n}$ is not simple in the finite characteristic case, still any homomorphism of $A_{n}$ to itself must be an embedding. It turns out that situation is not as simple as he conjectured and though it is impossible to have a homomorphism of $A_{n}$ into $A_{m}$ if $m< n$ there are homomorphism of $A_{n}$ into $A_{n}$ with rather large kernels.

Maps preserving $xy*=0$ (Talk slides)

Abstract. Let $R$ be a prime ring with symmetric Martindale quotient ring $Q_s(R)$. Assume that $R$ has an involution $^*$ and contain nontrivial idempotents. In this paper we prove that if $R$ is a prime ring with involution $^*$, containing nontrivial idempotents and $\delta : R \longrightarrow R$ is an additive map such that $\delta (x)y^* +x\delta(y)^*=0$ whenever $xy^*=0$, then there exists a $^*$-derivation $g : Q \longrightarrow Q$ such that $\delta(xy)=\delta(x)y+xg(y)$ for all $x, y \in R$.

Quasi-Baer ring hulls and their applications (Talk slides)

Abstract. A ring $R$ is called quasi-Baer if the right annihilator of each two-sided ideal is generated by an idempotent. The origin of the notion of quasi-Baer rings goes back to the characterization of a finite dimensional algebra over an algebraically closed field by W. Clark. We discuss the existence and uniqueness of a quasi-Baer ring hull of a semiprime ring. Also we discuss strong connections between FI-extending (i.e., fully invariant extending) and the quasi-Baer properties for a ring. Applications of quasi-Baer ring hulls to investigate boundedly centrally closed $C^{*}$-algebras and extended centroids of $C^{*}$-algebras are discussed. (This is a joint work with Gary F. Birkenmeier and S. Tariq Rizvi)

Some questions and results related to Koethe's nil ideal problem (Talk slides)

Abstract. It is easy to check that the sum of any family of two-sided nil ideals of an associative ring is a nil ideal as well. Does the same hold for left nil ideals? Though this question looks very elementary and was raised more than seventy years ago (in 1930 by Koethe) it is still open. It is called Koethe's nil ideal problem and is one of the most famous open problems in ring theory. Attempts to solve it led to many interesting, deep and sometimes surprising results. There are also many related open problems. The aim of the talk is to present several such problems as well as some old and new results obtained in the area. A particular attention will be paid to results obtained by Professor P.H. Lee and his collaborators.

Recent developments in the theory of Baer and Rickart modules (Talk slides)

Abstract. A ring is called Baer (right Rickart) if the right annihilator of any subset (single element) of $R$ is generated by an idempotent of $R$.
 
Using the endomorphism ring of a module, we extended these two notions to a general module theoretic setting, recently: Let $R$ be any ring, $M$ be an $R$-module and $S =\text{End}_R(M)$. $M$ is said to be a Baer module if the right annihilator in $M$ of any subset of $S$ is generated by an idempotent of $S$. Equivalently, the left annihilator in $S$ of any submodule of $M$ is generated by an idempotent of $S$. The module $M$ is called a Rickart module if the right annihilator in $M$ of any single element of $S$ is generated by an idempotent of $S$, equivalently, $r_M(\varphi)=\text{Ker} \varphi \leq^\oplus M$ for every $\varphi \in S$. In this talk we will compare and contrast the two notions and present their properties. Endomorphism rings of these modules and their direct sums will be discussed. We will present some recent developments in this theory including a dual notion.

An algebraic approach to tropical mathematics (Talk slides)

Abstract. The rapidly developing topic called "tropical mathematics," has been based on two main approaches. Primarily, tropical curves have been defined as domains of non-differentiability of polynomials over the max-plus algebra, and also tropical mathematics has been viewed in terms of valuation theory applied to curves over Puiseux series. Unfortunately, semirings such as the max-plus algebra possess a limited algebraic structure theory, and also do not reflect these valuation-theoretic properties, thereby forcing researchers to turn to combinatoric arguments.
 
The object of this talk is to present an algebraic structure more compatible with algebraic structure theory and valuation theory than the max-plus algebra. We present a "layered" structure, "sorted" by a semiring which permits varying ghost layers, and indicate how it permits a direct algebraic description of tropical varieties. We also discuss factorization of polynomials, linear algebra, properties of the resultant, and multiple roots of polynomials. Explicit examples and comparisons are given for various "sorting" semirings such as the natural numbers and the positive rational numbers, and we consider how this theory relates to some recent developments in the tropical literature such as "characteristic 1," "analytification," and "hyperfields."

Symmetries on bounded observables (Talk slides)

Abstract. In mathematical foundations of quantum mechanics bounded observables are represented by self-adjoint operators on a Hilbert space. Various operations and relations on self-adjoint operators are important because of applications in mathematical physics. Bijective maps on self-adjoint operators preserving such operations or relations are called symmetries. Some recent results on such maps will be presented.

A matrix representation of an Azumaya group ring (Talk slides)

Abstract. Let $R$ be an indecomposable ring with $1$ of characteristic $p^k$ for some prime integer $p$ and integer $k$, $G$ a group, and $RG$ a group ring of $G$ over $R$. It is shown that if $RG$ is an Azumaya algebra, then $RG$ contains a direct sum of matrix rings over Azumaya algebras, and $RG$ is a direct sum of matrix rings over Azumaya algebras if and only if the center of $RCG'$ is $C$ where $C$ is the center of $RG$ and $G'$ is the commutator subring of $G$.

A Jordan derivation of a generalized matrix algebra need not be a derivation (Talk slides)

Abstract. In this talk all algebras, rings and modules are assumed to be $2$-torsion free. Every derivation from an algebra $A$ to an $A$-module $M$ is a Jordan derivation, but the converse is not true, as was shown by Benkovic in 2005. Herstein proved in 1957 that every Jordan derivation of a prime ring, i.e., every Jordan derivation from a prime ring to itself, is a derivation. Bresar showed in 1988 that Herstein's result is true for semiprime rings. Herstein's result was also proved in various other settings, by amongst others, Sinclair in 1970 for semisimple Banach algebras, and by Zhang in 1998 for nest algebras. Zhang also proved in 2006 that every Jordan derivation of a triangular matrix algebra is a derivation. In this talk we give a complete description of all the Jordan derivations of a generalized matrix algebra (sometimes also called a Morita context ring associated with a Morita context), and we give an example of a Jordan derivation of such an algebra which is not a derivation.

On Frobenius algebras and Frobenius functors (Talk slides)

Abstract. Frobenius algebras are defined as finite dimensional algebras $A$ over a field $K$ which allow for a non-degenerated associative bilinear form $\beta:A\times A\to K$ - obviously a notion from linear algebra. It was shown by L. Abrams (in 1999) that any Frobenius algebra has a structure of a coalgebra such that the category of $A$-comodules is equivalent to the category $A$-modules.
 
The purpose of the talk is to interprete all these properties from the categorical point of view. This leads to the notion of Frobenius functors and Frobenius monads defined on arbitrary categories and it shows that Abrams' result is obvious from an observation on adjoint functors made in a paper by S. Eilenberg and J.C. Moore (in 1965).

The range and kernel inclusion of algebraic derivations

Abstract. Let $\delta$ and $\delta'$ be derivations of a prime ring $R$ and let $\delta$ be algebraic. In this talk, I shall explain the relation between the following two conditions (i) the range of $\delta'$ is contained in the range of $\delta$, (ii) the kernel of $\delta$ is contained in the kernel of $\delta'$. (This is a joint work with Chen-Lian Chuang and Tsiu-Kwen Lee).

An application of a theorem of clean endomorphism rings (Talk slides)

Abstract. A theorem of Camillo, Khurana, Lam, Nicholson and Zhou in [Continuous modules are clean, J. Alg. 304(1)(2006), 94-111] states that every endomorphism of a continuous module is the sum of an automorphism and an idempotent endomorphism. This was proved using a so-called ABCD-decomposition of a clean endomorphism. In this talk, a more direct proof of this theorem will be outlined, and an application to the study of quasi-injective modules will be given, which leads to solving Dinh's open question whether a pseudo-injective CS module is quasi-injective. (Based on joint work with V.P. Camillo, D. Khurana, T.Y. Lam and W.K. Nicholson, and joint work with Tsiu-Kwen Lee).

Right Gaussian rings relative to a monoid (Talk slides)

Abstract. In this talk we introduce a class of rings we call right Gaussian rings, defined by the property that for any two polynomials $f$, $g$ over the ring $R$, the right ideal of R generated by the coefficients of the product $fg$ coincides with the product of the right ideals generated by the coefficients of $f$ and that of $g$, respectively. Prüfer domains are precisely commutative domains belonging to this new class of rings. In this talk we adduce the connections between right Gaussian rings and the classes of Armendariz rings and rings whose right ideals form a distributive lattice. We characterize skew power series rings that are right Gaussian, extending to the noncommutative case a well-known result by Anderson and Camillo. We also study quotient rings of right Gaussian rings. (This is a joint work with Ryszard Mazurek.)