Topics in ring theory.

*(English)*Zbl 0232.16001
Chicago Lectures in Mathematics. Chicago-London: The University of Chicago Press. xi, 132 p. £1.15 (1969).

[These notes are essentially the publication in (paperback) book form of the author’s 1965 University of Chicago Mathematics Lecture Notes publication of the same title.]

This book contains many interesting and useful results in non-commutative ring theory. These results require methods which are primarily computational and which involve internal structure rather than being module-theoretic in nature.

Chapter 1 contains results of W. Baxter and I. N. Herstein on the structure of Jordan and Lie ideals in simple associative rings. For example, if \(R\) is a simple ring and \(\mathrm{char } R\ne 2\), then \(R\) is simple as a Jordan ring with the product \(x\cdot y= xy+yx\); with respect to the Lie product \([x,y]=xy-yx\), the Lie ideals of \(R\) are either contained in the center of \(R\) or contain \([R,R]\); the proper Lie ideals of \([R,R]\) are in the center of \(R\). This structure theory for Lie ideals is used to obtain a result of S. A. Amitsur on invariant subspaces of simple rings.

Chapter 2 deals with simple rings with involution whose characteristic is not 2. The results of Chapter 1 are used to show such things as: the set \(S\) of symmetric elements is simple as a Jordan ring; the Lie ideals of the set \(K\) of skew-symmetric elements are either contained in the center of \(R\), or contain \([K,K]\), provided that the dimension of \(R\) over its center is greater than 16; the proper Lie ideals of \([K,K]\) are in the center of \(R\), provided that the dimension of \(R\) exceeds 16 over its center. Once again, these results are due to W. Baxter and I. N. Herstein.

Chapter 3 presents material on mappings of the Jordan structure of rings. Two of the major results are that Jordan derivations of prime rings are derivations, and that a Jordan homomorphism onto a prime ring is either a homomorphism or an anti-homomorphism. Some related and more recent results due to W. Martindale are also discussed in this chapter.

Chapter 4 contains the fundamental theorems of A. W. Goldie about rings satisfying certain ascending chain conditions. The proofs given are due to I. N. Herstein and to C. Procesi and L. Small. Chapters 5 and 6 continue the study of rings with chain conditions. The first half of Chapter 5 is concerned with the effect of various chain conditions on nil rings. The goal is to show that the ring is either locally nilpotent, or nilpotent. The remainder of Chapter 5 is a proof of a theorem of E. Posner about the structure of a prime ring satisfying a polynomial identity. Chapter 6 contains material due largely to L. Lesieur and R. Croisot, and is, in the author’s words, “…an attempt to simulate in a non-commutative Noetherian ring the Lasker-Noether primary decomposition in a commutative Noetherian ring.”

Chapter 7 concludes the book with the presentation of some recent counter-examples. Included is the Golod-Shafarevich theorem, the example of G. Bergman of a right but not left primitive ring, and the example of E. Sasiada of a simple radical ring.

This book contains many interesting and useful results in non-commutative ring theory. These results require methods which are primarily computational and which involve internal structure rather than being module-theoretic in nature.

Chapter 1 contains results of W. Baxter and I. N. Herstein on the structure of Jordan and Lie ideals in simple associative rings. For example, if \(R\) is a simple ring and \(\mathrm{char } R\ne 2\), then \(R\) is simple as a Jordan ring with the product \(x\cdot y= xy+yx\); with respect to the Lie product \([x,y]=xy-yx\), the Lie ideals of \(R\) are either contained in the center of \(R\) or contain \([R,R]\); the proper Lie ideals of \([R,R]\) are in the center of \(R\). This structure theory for Lie ideals is used to obtain a result of S. A. Amitsur on invariant subspaces of simple rings.

Chapter 2 deals with simple rings with involution whose characteristic is not 2. The results of Chapter 1 are used to show such things as: the set \(S\) of symmetric elements is simple as a Jordan ring; the Lie ideals of the set \(K\) of skew-symmetric elements are either contained in the center of \(R\), or contain \([K,K]\), provided that the dimension of \(R\) over its center is greater than 16; the proper Lie ideals of \([K,K]\) are in the center of \(R\), provided that the dimension of \(R\) exceeds 16 over its center. Once again, these results are due to W. Baxter and I. N. Herstein.

Chapter 3 presents material on mappings of the Jordan structure of rings. Two of the major results are that Jordan derivations of prime rings are derivations, and that a Jordan homomorphism onto a prime ring is either a homomorphism or an anti-homomorphism. Some related and more recent results due to W. Martindale are also discussed in this chapter.

Chapter 4 contains the fundamental theorems of A. W. Goldie about rings satisfying certain ascending chain conditions. The proofs given are due to I. N. Herstein and to C. Procesi and L. Small. Chapters 5 and 6 continue the study of rings with chain conditions. The first half of Chapter 5 is concerned with the effect of various chain conditions on nil rings. The goal is to show that the ring is either locally nilpotent, or nilpotent. The remainder of Chapter 5 is a proof of a theorem of E. Posner about the structure of a prime ring satisfying a polynomial identity. Chapter 6 contains material due largely to L. Lesieur and R. Croisot, and is, in the author’s words, “…an attempt to simulate in a non-commutative Noetherian ring the Lasker-Noether primary decomposition in a commutative Noetherian ring.”

Chapter 7 concludes the book with the presentation of some recent counter-examples. Included is the Golod-Shafarevich theorem, the example of G. Bergman of a right but not left primitive ring, and the example of E. Sasiada of a simple radical ring.

Reviewer: Charles Lanski (Los Angeles)

##### MSC:

16-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to associative rings and algebras |

16D30 | Infinite-dimensional simple rings (except as in 16Kxx) |

16N60 | Prime and semiprime associative rings |

16N40 | Nil and nilpotent radicals, sets, ideals, associative rings |

17C50 | Jordan structures associated with other structures |

16P60 | Chain conditions on annihilators and summands: Goldie-type conditions |

16P40 | Noetherian rings and modules (associative rings and algebras) |

16R20 | Semiprime p.i. rings, rings embeddable in matrices over commutative rings |