By Donald S. Passman
First released in 1991, this ebook includes the middle fabric for an undergraduate first direction in ring idea. utilizing the underlying subject matter of projective and injective modules, the writer touches upon numerous facets of commutative and noncommutative ring conception. particularly, a few significant effects are highlighted and proved. the 1st a part of the booklet, known as "Projective Modules", starts off with easy module concept after which proceeds to surveying quite a few exact sessions of earrings (Wedderburn, Artinian and Noetherian jewelry, hereditary earrings, Dedekind domain names, etc.). This half concludes with an advent and dialogue of the thoughts of the projective size. half II, "Polynomial Rings", reports those jewelry in a mildly noncommutative environment. the various effects proved comprise the Hilbert Syzygy Theorem (in the commutative case) and the Hilbert Nullstellensatz (for virtually commutative rings). half III, "Injective Modules", comprises, particularly, quite a few notions of the hoop of quotients, the Goldie Theorems, and the characterization of the injective modules over Noetherian jewelry. The booklet comprises a number of workouts and an inventory of recommended extra studying. it truly is appropriate for graduate scholars and researchers drawn to ring concept.
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Extra resources for A Course in Ring Theory (AMS Chelsea Publishing)
Fix a lattice ✘ ✞ ✥ ⑧ and let ❁❃✤ (a) Let ✂ ✦ ✄ ✘ be finite. Prove that ✆✧✣✦☞✩★ ✥❯✂ ✭✯③❲④ ⑤✯❁❃✪❂ ✣✥☞✺✤❋✂❇✬ . (b) Prove that ✤ (c) Prove that ✤ is the largest subspace contained in . Chapter 1. Affine Toric Varieties 34 ✂ ✄ ✠ ③❵④ ⑤✁ ✦☞❵❄ ✤❋✂✮✬✮✭✯③❲④⑥⑤❱❁❲❂✤✣✥☞❡✤ ✂✮✬ ☎ (b) Let ✦✘ be finite. Prove that . 18. 7. 20. 8. 21. Hint: First show that when a cone is smooth, the ray generators of the cone also generate the corresponding semigroup. 21 as a union of such cones. 10. Let ✂❱✄ ✞✜✠ be a cone generated by a set of linearly independent vectors in ✞ ✠ .
Chapter 1. Affine Toric Varieties 16 ✜✧ be a sublattice. 10. Let ✥✑✦ (a) The ideal ✻✣ ❘ ✟ ✳ ✥✍☛ is called a lattice ideal. (b) A prime lattice ideal is called a toric ideal. 9 are toric ideals. ) In each example, we have a prime ideal generated by binomials. As we now show, such ideals are automatically toric. 11. An ideal and generated by binomials. ✒✞ ✻ ✦ ✟❋✪ ✬✜✭✍✁✔✓✔✓✔✓☞✁✯✬ ✧ ✲ is toric if and only if it is prime ✻ ✎✞ Proof. One direction is obvious. So suppose that is prime and generated by ✬ ❘ ✬ .
We next study the question of when an affine toric ❀ variety is normal. We need one definition before stating our normality criterion. 4. An affine semigroup ✷ ✷ ✣ and ✱✗✳ ❀ , ✱✗✳ implies ✱✗✳ . 4). 5. Let are equivalent: (a) ❀ be an affine toric variety with torus .
A Course in Ring Theory (AMS Chelsea Publishing) by Donald S. Passman