| Preface to the Revised Edition | p. xiii |
| Preface | p. xv |
| Notation | p. xix |
| Preliminaries | p. 1 |
| Chain Conditions | p. 1 |
| Prime Radical | p. 4 |
| Jacobson Radical | p. 6 |
| Basic Theory | |
| Some Noetherian Rings | p. 11 |
| Matrices | p. 11 |
| Skew Polynomial Rings | p. 15 |
| Weyl Algebras | p. 19 |
| Skew Power Series and Laurent Polynomials | p. 21 |
| Group Rings and Generalizations | p. 22 |
| Skew Polynomial Rings in Several Variables | p. 26 |
| Enveloping Algebras and Their Generalizations | p. 30 |
| Further Simple Rings | p. 35 |
| Additional Remarks | p. 38 |
| Quotient Rings and Goldie's Theorem | p. 41 |
| Right Quotient Rings | p. 41 |
| Uniform Dimension | p. 51 |
| Goldie's Theorem | p. 57 |
| Additional Remarks | p. 62 |
| Structure of Semiprime Goldie Rings | p. 63 |
| Orders in Quotient Rings | p. 63 |
| Minimal Primes | p. 70 |
| Right Ideals in Semiprime Rings | p. 74 |
| Endomorphism Rings | p. 78 |
| Morita Equivalence | p. 83 |
| Morita Contexts and Prime Rings | p. 89 |
| Additional Remarks | p. 94 |
| Semiprime Ideals in Noetherian Rings | p. 97 |
| Reduced Rank and Applications | p. 98 |
| Artin-Rees Property and Localization | p. 106 |
| Localization and Prime Ideals | p. 112 |
| Affiliated Primes and Regular Elements | p. 123 |
| Additivity | p. 128 |
| Patch Continuity | p. 131 |
| Additional Remarks | p. 138 |
| Some Dedekind-like Rings | p. 141 |
| Maximal Orders | p. 142 |
| Asano and Dedekind Prime Rings | p. 146 |
| Classical Orders | p. 152 |
| Hereditary Noetherian Rings | p. 159 |
| Idealizer Rings | p. 162 |
| Hereditary Noetherian Prime Rings | p. 166 |
| Modules over Dedekind Prime Rings | p. 172 |
| Additional Remarks | p. 180 |
| Dimensions | |
| Krull Dimension | p. 185 |
| Deviation of a Poset | p. 186 |
| Krull Dimension of Modules | p. 192 |
| Krull Dimension in Rings | p. 200 |
| Prime Ideals and FBN Rings | p. 204 |
| Bounds on Krull Dimensions | p. 209 |
| Calculation of Krull Dimension | p. 213 |
| Stable Bounds on Generators | p. 219 |
| Quotient Rings and Localization | p. 223 |
| Skew Polynomials over Commutative Rings | p. 233 |
| Additional Remarks | p. 242 |
| Global Dimension | p. 245 |
| Preliminaries | p. 245 |
| Change of Rings | p. 250 |
| Factor Rings | p. 253 |
| Localization | p. 260 |
| Estimates of Global Dimension | p. 261 |
| Filtered and Graded Modules | p. 270 |
| Regular Rings | p. 275 |
| Fixed Rings | p. 278 |
| Skew Polynomial Rings | p. 282 |
| Skew Polynomials over Commutative Rings | p. 290 |
| Simple Dedekind Domains | p. 291 |
| Additional Remarks | p. 294 |
| Gelfand-Kirillov Dimension | p. 297 |
| Definition and Examples | p. 298 |
| Dimensions of Related Algebras | p. 304 |
| Module Theory | p. 311 |
| Almost Commutative Algebras and Hilbert Polynomials | p. 320 |
| Applications | p. 324 |
| Somewhat Commutative Algebras | p. 330 |
| Additional Remarks | p. 337 |
| Extensions | |
| The Nullstellensatz | p. 341 |
| Algebras over a Field | p. 341 |
| Algebras over Rings | p. 346 |
| Generic Flatness | p. 348 |
| Constructible Algebras | p. 351 |
| Finite Dimensional Endomorphism Rings | p. 356 |
| Polynomials over Division Rings | p. 358 |
| Additional Remarks | p. 362 |
| Prime Ideals in Extension Rings | p. 365 |
| Finite Extensions and Chain Conditions | p. 366 |
| Prime Ideals | p. 370 |
| Quotient Rings and Closure | p. 375 |
| Incomparability | p. 380 |
| Crossed Products and Fixed Rings | p. 387 |
| Primes in Polynomial Rings | p. 392 |
| Additional Remarks | p. 400 |
| Stability | p. 403 |
| Stably Free Modules | p. 404 |
| Stably Free Nonfree Modules | p. 409 |
| Stable and Elementary Ranks | p. 413 |
| Cancellation of Modules | p. 418 |
| Ranks of Certain Rings | p. 421 |
| Local Information | p. 428 |
| Stability and Cancellation | p. 435 |
| Additional Remarks | p. 443 |
| K[subscript 0] and Extension Rings | p. 445 |
| K[subscript 0] of a Ring | p. 446 |
| Projective-Graded Modules | p. 449 |
| Filtrations and the Syzygy Theorem | p. 453 |
| K[subscript 0] of Module Categories | p. 456 |
| Skew Laurent Extensions | p. 461 |
| Filtered Rings | p. 463 |
| Applications to Simple Rings | p. 467 |
| Additional Remarks | p. 472 |
| Examples | |
| Polynomial Identity Rings | p. 477 |
| Polynomial Identities | p. 478 |
| Nilpotence | p. 484 |
| Central Simple Algebras | p. 486 |
| Embeddings and Matrix Rings | p. 489 |
| Central Polynomials | p. 492 |
| Semiprime Rings and Central Polynomials | p. 497 |
| Prime Ideals and Azumaya Algebras | p. 501 |
| Integral Extensions and Prime Rings | p. 507 |
| The Trace Ring and Maximal Orders | p. 514 |
| Affine [kappa]-algebras | p. 518 |
| Additional Remarks | p. 523 |
| Enveloping Algebras of Lie Algebras | p. 525 |
| Basics | p. 526 |
| Prime Ideals | p. 530 |
| Eigenvalues and Prime Ideals | p. 534 |
| Primitive Ideals | p. 537 |
| The Solvable Case | p. 542 |
| When Eigenvectors are Central | p. 545 |
| When g is Algebraic | p. 550 |
| The Simple Algebras A(V, [delta], [Gamma]) | p. 552 |
| The General Case | p. 557 |
| Additional Remarks | p. 563 |
| Rings of Differential Operators on Algebraic Varieties | p. 565 |
| Algebras over a Ring | p. 566 |
| Affine Algebras over a Field | p. 573 |
| Dimensions | p. 578 |
| Further Properties | p. 585 |
| Rings of Differential Operators | p. 590 |
| Additional Remarks | p. 593 |
| References | p. 595 |
| Index of Notation | p. 621 |
| Index | p. 625 |
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