This text, drawn from the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-term course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semi-simple rings, Jacobson's theory of the radical representation theory of groups and algebras, prime and semi-prime rings, primitive and semi-primitive rings, division rings, ordered rings, local and semi-local rings, and perfect and semi-perfect rings. By aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation, the author has produced a text which is suitable not only for use in a graduate course, but also for self-study by other interested graduate students. Numerous exercises are also included. This graduate textbook on rings, fields and algebras is intended for graduate students in mathematics.

A first course in noncommutative rings, by T. Y. Lam. Pp. 385. £37 (pb), £62.50 (hb). 2001. ISBN 0 387 95325 6 (pb), 0 387 95183 0 (hb) (Springer-Verlag).

Determine the value of the variable in each equation.

Basic Algebra = 代数学入門

We generalize the results of Leger and Luks about generalized derivations of Lie algebras to the case of color $n$-ary $\Omega$-algebras. Particularly, we prove some properties of generalized derivations of color $n$-ary algebras; prove that a quasiderivation algebra of a color $n$-ary $\Omega$-algebra can be embedded into the derivation algebra of a larger color $n$-ary $\Omega$-algebra, and describe (anti)commutative $n$-ary algebras satisfying the condition $QDer = End.$

Generalized derivations of (color) n-ary algebras

We generalize the results of Leger and Luks, Zhang R. and Zhang Y.; Chen, Ma, Ni, Niu, Zhou and Fan; Kaygorodov and Popov about generalized derivations of color n-ary algebras to the case of n-ary Hom-
 $$\Omega $$
 color algebras. Particularly, we prove some properties of generalized derivations of multiplicative n-ary Hom-
 $$\Omega $$
 color algebras. Moreover, we prove that the quasiderivation algebra of any multiplicative n-ary Hom-
 $$\Omega $$
 color algebra can be embedded into the derivation algebra of a larger multiplicative n-ary Hom-
 $$\Omega $$
 color algebra.

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Generalized Derivations of Multiplicative $n$-Ary Hom-$\Omega$ Color Algebras

Let A be a unital algebra over a field with char, and let be linear maps. We say that f is a -derivation if for all , and we say that f is a Jordan -derivation if for all (here, ). We show that if the property that every Jordan -derivation is a -derivation holds in A, then so does in the algebra for every commutative unital algebra S. We also show that every semiprime algebra A has this property. Combining these two results, it follows, in particular, that the classical Jordan derivations are derivations on the tensor product between a semiprime and a commutative algebra.

Jordan {g, h}-derivations on tensor products of algebras

Let R be a prime ring which is not commutative, with maximal symmetric ring of quotients Q ms (R), and let τ be an anti-automorphism of R. An additive map δ: R → Q ms (R) is called a Jordan τ-derivation if δ(x 2) = δ(x)x τ + xδ(x) for all x ∈ R. A Jordan τ-derivation of R is called X-inner if it is of the form x → ax τ − xa for x ∈ R, where a ∈ Q ms (R). It is proved that any Jordan τ-derivation of R is X-inner if either R is not a GPI-ring or R is a PI-ring except when charR = 2 and dim  C RC = 4, where C is the extended centroid of R.

Jordan τ-derivations of Prime Rings

Jordan derivations of prime rings with characteristic two

Let D be a division algebra over an infinite field K such that every element of D is a sum of finitely many algebraic elements. As a generalization of Asano’s theorem, it is proved that every nonce...

On Asano’s theorem

Outer inverses, minus partial orders, and triplet invariance

Let R be a semiprime ring with τ an anti-endomorphism or endomorphism. It is proved that if τ satisfies an Engel condition [[…[[xτ,xn1],xn2]…],xnk]=0 for all x∈R, where n1,n2,…,nk are k fixed positive integers, then τ is a commuting map (i.e. [xτ,x]=0 for all x∈R). The theorem generalizes the results proved in [22] with τ an anti-automorphism of finite order and in [10, 11] with R a division ring, respectively.

Jheng-Huei Lin

Papers

Jordan τ-derivations of Prime Rings

Jordan derivations of prime rings with characteristic two

On Asano’s theorem

Outer inverses, minus partial orders, and triplet invariance

Anti-endomorphisms and endomorphisms satisfying an Engel condition