Exercises in Classical Ring Theory pptx

[...]... 6 Exercises §16 Polynomials over Division Rings 18 Exercises 201 201 161 178 211 228 231 6 Ordered Structures in Rings 247 §17 Orderings and Preorderings in Rings 247 15 Exercises §18 Ordered Division Rings 258 7 Exercises 7 Local Rings, Semilocal Rings, and Idempotents §19 Local Rings... semiprimitive rings, division rings, ordered rings, local and semilocal rings, and the theory of idempotents, and ending with perfect and semiperfect rings For the reader’s information, we should note that this book does not include problems in the vast areas of module theory (e.g., projectivity, injectivity, and flatness), category theory (e.g., equivalences and dualities), or rings of quotients (e.g., Ore rings... researchers in ring theory and other closely allied fields This book is organized in the same way as FC, in eight chapters and twenty-five sections It deals mainly with the classical parts of ring theory, starting with the Wedderburn-Artin theory of semisimple rings, Jacobson’s theory of the radical, and the representation theory of groups and algebras, then continuing with prime and semiprime rings, primitive... This page intentionally left blank Chapter 1 Wedderburn-Artin Theory §1 Basic Terminology and Examples The exercises in this beginning section cover the basic aspects of rings, ideals (both 1-sided and 2-sided), zero-divisors and units, isomorphisms of modules and rings, the chain conditions, and Dedekind-finiteness A ring R is said to be Dedekind-finite if ab = 1 in R implies that ba = 1 The chain conditions... pertaining to direct decompositions of a ring into 1-sided or 2-sided ideals (Exercises 7 and 8) Throughout these exercises, the word ring means an associative (but not necessarily commutative) ring with an identity element 1 (On a few isolated occasions, we shall deal with rings without an identity.1 Whenever this happens, it will be clearly stated.) The word “subring” always means a subring containing... appearing (or to appear) in the same section, we shall sometimes drop the section number from the reference Thus, when we refer to “Exercise 7” within §12, we shall mean Exercise 12.7 The ring theory conventions used in this book are the same as those introduced in FC Thus, a ring R means a ring with identity (unless otherwise specified) A subring of R means a subring containing the identity of R (unless... the ring k polynomial ring over k with (commuting) variables {xi : i ∈ I} free ring over k generated by {xi : i ∈ I} skew polynomial ring with respect to an endomorphism σ on k differential polynomial ring with respect to a derivation δ on k commutator subgroup of the group G additive subgroup of the ring R generated by all [a, b] = ab − ba finitely generated ascending chain condition descending chain... (ACC) or artinian (DCC) conditions which can be imposed on submodules of a module, or on 1-sided or 2-sided ideals of a ring Some of the exercises in this section lie at the foundations of noncommutative ring theory, and will be used freely in all later exercises These include, for instance, the computation of the center of a matrix ring (Exercise 9), the computation of the endomorphism ring for n (identical)... algebraic algebra over k For other interesting properties of such an algebra, see Exercises 12.6B, 13.11, and 23.6(2) Ex 1.14 (Kaplansky) Suppose an element a in a ring has a right inverse b but no left inverse Show that a has in nitely many right inverses (In particular, if a ring is finite, it must be Dedekind-finite.) Solution Suppose we have already constructed n distinct right inverses b1 , , bn for a... and Semiprime Rings 32 Exercises §11 Structure of Primitive Rings; the Density Theorem 24 Exercises §12 Subdirect Products and Commutativity Theorems 32 Exercises 141 141 Introduction to Division Rings §13 Division Rings 19 Exercises §14 Some Classical Constructions 19 Exercises §15 . Structures in Rings 247 §17. Orderings and Preorderings in Rings 247 15 Exercises §18. Ordered Division Rings 258 7 Exercises 7 Local Rings, Semilocal Rings,. After writing A First Course in Noncommutative Rings (Springer-Verlag GTM 131, hereafter referred to as “FC ”), I taught ring theory in Berkeley again in the