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).

Nil clean rings

Натрийуретические пептиды (НУП) являются важными биомаркерами в диагностике и определении прогноза у пациентов с сердечной недостаточностью (СН). Оценка динамики концентрации НУП (BNP, Nt -proBNP) может быть использована в качестве критерия успешности проводимой терапии. так, при достижении целевых уровней НУП можно прогнозировать благоприятный исход заболевания. В настоящее время лечение СН с учетом уровней НУП является частью рекомендаций по лечению СН (класс IIа) и улучшению ее исхода (класс IIб) в США, однако такой подход не используется в российских клиниках. Цель. Представить современный взгляд на возможность использования НУП для оценки эффективности проводимой терапии пациентов с СН. Ключевые слова: натрийуретические пептиды, сердечная недостаточность, оценка эффективности терапии.

Федеральное государственное бюджетное учреждение «Научно-исследовательский институт комплексных проблем сердечно-сосудистых заболеваний» Сибирского отделения Российской академии медицинских наук, Кемерово, Россия

Nil-clean and strongly nil-clean rings

Nil-clean matrix rings

We answer multiple open questions concerning lifting of idempotents that appear in the literature. Most of the results are obtained by constructing explicit counter-examples. For instance, we provide a ring R for which idempotents lift modulo the Jacobson radical J(R), but idempotents do not lift modulo J(&#x1d544;2(R)). Thus, the property “idempotents lift modulo the Jacobson radical” is not a Morita invariant. We also prove that if I and J are ideals of R for which idempotents lift (even strongly), then it can be the case that idempotents do not lift over I + J. On the positive side, if I and J are enabling ideals in R, then I + J is also an enabling ideal. We show that if I ⊴ R is (weakly) enabling in R, then I[t] is not necessarily (weakly) enabling in R[t] while I⟦t⟧ is (weakly) enabling in R⟦t⟧. The latter result is a special case of a more general theorem about completions. Finally, we give examples showing that conjugate idempotents are not necessarily related by a string of perspectivities.

Idempotent lifting and ring extensions

Strongly clean matrices over arbitrary rings

A note on completeness and strongly clean rings

In this article we investigate the annihilating-ideal graph of a commutative ring, introduced by Behboodi and Rakeei in [10]. Our main goal is to determine which algebraic properties of a ring are ...

/pdf/classifying-annihilating-ideal-graphs-of-commutative-513dz1nobz.pdf

Alexander J. Diesl

Papers

Nil clean rings

Idempotent lifting and ring extensions

Strongly clean matrices over arbitrary rings

A note on completeness and strongly clean rings

Classifying annihilating-ideal graphs of commutative artinian rings