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).

A semigroup S is called E-inversive if for every a ∈ S there exists x ∈ S such that ax is idempotent. S is called E-semigroup if the set of idempotents of S forms a subsemigroup. In this paper some special congruences on E-inversive E-semigroups are investigated, such as the least group congruence, a certain semilattice congruence, some regular congruences and a certain idempotent-separating congruence.

Certain congruences on E-inversive E-semigroups

the Pentagon were not only a human tragedy but also had potentially serious ramifications for the economy. The most immediate economic effect of the attacks was a temporary inability to clear checks, caused by the suspension of flights. In addition, the New York Stock Exchange and other financial markets closed for the rest of the week. Many individuals, perhaps fearful of further attacks, withdrew money from the bank. Similarly, businesses moved money from illiquid assets, like stocks and bonds, to liquid assets, like checking accounts. In the medium term, the attacks have reduced consumption and investment through their effects on consumer confidence and the temporary fall in stock prices. This reduction in consumption and investment exacerbated the economic slowdown that was already developing. In the month after the attack, for example, the Blue Chip Consensus GDP growth forecast for 2001 was revised down from 1.6 percent to 1.1 percent, and the same figure for 2002 fell from 2.7 percent to 1.5 percent. In the longer term, resources will shift to law enforcement and defense, away from the production of other goods and services. As law enforcement and defense are mostly provided by the government, taxes will rise somewhat to pay for them. Also, many of the productive resources destroyed in the attacks will be replaced. The size of these effects can easily be overstated, however. Compared to the whole U.S. economy , the increased security outlays will be very small. The attacks will also shift resources among industries in the longer term by raising the costs of activities like travel, postal services, security and insurance. That is, travelers will require more security to fly, and firms will have to pay more for a given level of property insurance for a downtown office building. These higher costs will reduce the quantity of travel, tourism and construction produced and consumed. The Federal Reserve's response to the immediate effects of the attacks was to provide liquidity—the ability to make payments—to firms and individuals. Particularly important was providing liquidity to financial firms, which constantly buy and sell assets, because they must make payments with either funds from recently sold assets or money borrowed from banks. During times of crisis, however, banks avoid making such loans because falling asset prices threaten the value of the collateral. An interruption in bank lending to financial firms could potentially set off a domino chain of bankruptcies that would …

五角大楼（The Pentagon）

Magnifying elements in semigroups

https://www-users.york.ac.uk/~jbf1/covers.pdf

Covers for monoids

Let X be a subset of a semigroup S We denote by E(X) the set of idempotent elements Of XAn element a of a semigroup S is called E-inverse if there exists such that  We note that the definition is not one-sided Indeed, an a element of a semigroup S is E-inversive if there exists such that (see [7], [l] p 98) A semigroup S is called E-inversive if all its elements are E-inversive This class of semigroups is extensive All semigroups with a zero and all eventually regular semigroups [2] are E-inversive semigroupsRecently E-inversive semigroups reappeared in a paper by Hall and Munn [3] and in a paper by Mitsch [5] The special case of E-inversive semigroups with pairwise commuting idempotents, called E-dense, was considered by Margolis and Pin [4] Let S and T be semigroups, and let be a homomorphism of S into the endomorphism semigroup of T If and , denote  Thus, if and then  The semidirect product of S and T ,in that order, with strutture map (Y, consists of the set S x T equipped with the product  This product will be denoted by S _{𝛼}T In this note we determine which semidirect products of semigroups are E-inversive semigroups and E-dense semigroups, respectively It turns out that the case in which S induces only automorphism on T allows a particularly simple description In [6], Preston has answered the analous question for regular semigroups and for inverse semigroups For the terminology and for the definitions of the algebraic theory of semigroups, we refer to [1]

On semidirect product of semigroups

Let M be a set. A set-theoretical solution of the pentagon equation on M is a map s:M×M→M×M such that s23 s13 s12=s12 s23, where s12=s×idM,s23=idM×s, and s13=(idM×τ)s12(idM×τ), and τ is the flip ma...

Set-theoretical solutions of the pentagon equation on groups

On some semigroups without increasing elements

This note contains a few introductory results on strongly Lie nilpotent rings and, in particular, an analogue of a well known theorem of P. Hall on nilpotent groups.

/pdf/note-on-strongly-lie-nilpotent-rings-4be4iid3yv.pdf

Francesco Catino

Papers

On semidirect product of semigroups

Set-theoretical solutions of the pentagon equation on groups

On some semigroups without increasing elements

Note on strongly Lie nilpotent rings

On a theorem of Gupta and Levin