## 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 |

CXLVI | 543 |

549 | |

553 | |

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algebra annihilator apply artinian ring assume called characterization clearly closed cogenerator commutative commutative ring complete condition consider construction contains Conversely Corollary course defined definition dimension direct direct sum division ring domain duality easy element embedded equivalent essential exact sequence example Exercise exists extended fact field finite flat Frobenius functor given gives Goldie hence holds homomorphism idempotent implies indecomposable injective instance isomorphism left ideal Lemma matrix maximal means Morita multiplication natural noetherian ring nonsingular nonzero notation Note notion particular prime principal projective Proof Proposition prove R-module rank regular relations Remark resp result right ideal right module right noetherian ring ring of quotients satisfies self-injective semiprime semisimple simple submodule subsection Suppose Theorem theory u.dim uniform write