## Lectures on Modules and RingsTextbook writing must be one of the cruelest of self-inflicted tortures. - Carl Faith Math Reviews 54: 5281 So why didn't I heed the warning of a wise colleague, especially one who is a great expert in the subject of modules and rings? The answer is simple: I did not learn about it until it was too late! My writing project in ring theory started in 1983 after I taught a year-long course in the subject at Berkeley. My original plan was to write up my lectures and publish them as a graduate text in a couple of years. My hopes of carrying out this plan on schedule were, however, quickly dashed as I began to realize how much material was at hand and how little time I had at my disposal. As the years went by, I added further material to my notes, and used them to teach different versions of the course. Eventually, I came to the realization that writing a single volume would not fully accomplish my original goal of giving a comprehensive treatment of basic ring theory. At the suggestion of Ulrike Schmickler-Hirzebruch, then Mathematics Editor of Springer-Verlag, I completed the first part of my project and published the write up in 1991 as A First Course in Noncommutative Rings, GTM 131, hereafter referred to as First Course (or simply FC). |

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### Contents

V | 1 |

VI | 2 |

VII | 5 |

VIII | 9 |

IX | 12 |

X | 16 |

XI | 17 |

XII | 21 |

LXXIX | 294 |

LXXX | 297 |

LXXXI | 298 |

LXXXII | 299 |

LXXXIII | 303 |

LXXXIV | 308 |

LXXXV | 314 |

LXXXVI | 317 |

XIII | 23 |

XIV | 30 |

XV | 34 |

XVI | 42 |

XVII | 45 |

XVIII | 48 |

XIX | 51 |

XX | 54 |

XXI | 60 |

XXIII | 64 |

XXIV | 69 |

XXV | 74 |

XXVI | 80 |

XXVII | 83 |

XXVIII | 90 |

XXIX | 96 |

XXX | 99 |

XXXI | 105 |

XXXII | 110 |

XXXIII | 113 |

XXXIV | 121 |

XXXV | 122 |

XXXVII | 127 |

XXXVIII | 129 |

XXXIX | 131 |

XL | 135 |

XLI | 136 |

XLII | 140 |

XLIII | 144 |

XLIV | 147 |

XLV | 153 |

XLVI | 159 |

XLVII | 165 |

XLIX | 173 |

L | 177 |

LI | 182 |

LII | 187 |

LIII | 192 |

LIV | 198 |

LV | 202 |

LVI | 207 |

LVII | 208 |

LIX | 214 |

LX | 219 |

LXI | 221 |

LXII | 228 |

LXIII | 232 |

LXIV | 236 |

LXV | 241 |

LXVI | 246 |

LXVII | 252 |

LXVIII | 253 |

LXIX | 260 |

LXX | 265 |

LXXI | 268 |

LXXII | 272 |

LXXIII | 275 |

LXXIV | 280 |

LXXV | 284 |

LXXVI | 287 |

LXXVII | 288 |

LXXVIII | 290 |

LXXXVII | 320 |

LXXXVIII | 323 |

LXXXIX | 331 |

XC | 334 |

XCI | 339 |

XCII | 342 |

XCIII | 345 |

XCIV | 347 |

XCV | 351 |

XCVI | 354 |

XCVII | 355 |

XCVIII | 357 |

XCIX | 358 |

C | 365 |

CI | 369 |

CII | 374 |

CIII | 380 |

CIV | 383 |

CV | 384 |

CVI | 389 |

CVII | 392 |

CVIII | 394 |

CIX | 401 |

CX | 403 |

CXI | 407 |

CXII | 408 |

CXIII | 412 |

CXIV | 414 |

CXV | 417 |

CXVI | 420 |

CXVII | 422 |

CXVIII | 427 |

CXIX | 431 |

CXX | 434 |

CXXI | 438 |

CXXII | 441 |

CXXIII | 450 |

CXXIV | 453 |

CXXV | 459 |

CXXVI | 461 |

CXXVIII | 470 |

CXXIX | 473 |

CXXX | 478 |

CXXXI | 480 |

CXXXII | 483 |

CXXXIII | 485 |

CXXXIV | 488 |

CXXXV | 490 |

CXXXVI | 496 |

CXXXVII | 501 |

CXXXVIII | 505 |

CXXXIX | 510 |

CXL | 515 |

CXLI | 518 |

CXLII | 522 |

CXLIII | 527 |

CXLIV | 534 |

CXLV | 537 |

543 | |

549 | |

553 | |

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### Common terms and phrases

abelian groups artinian ring assume automorphism bimodule commutative domain commutative noetherian commutative ring complement Corollary defined direct sum direct summand division ring embedded endomorphism example Exercise exists f.g. projective fact faithfully flat field finite-dimensional flat modules following are equivalent free module Frobenius algebra functor hence idempotent implies indecomposable injective hull injective module isomorphism left ideal left module Lemma matrix ring Morita duality nilpotent noetherian ring nonzero notation notion polynomial prime ideal progenerator projective module Proof Proposition prove QF ring Qr(R R-module r.gl.dim resp result Rickart right annihilators right artinian ring right module right noetherian ring right nonsingular right self-injective ring of quotients self-injective ring semihereditary semiprime ring semisimple ring short exact sequence simple right soc(RR stably finite submodule subring subsection surjection Theorem u.dim

### Popular passages

Page 548 - On continuous rings and self-injective rings, Trans. Amer. Math. Soc. 118(1965), 158-173.