By A. W. Chatters
The authors offer a concise advent to issues in commutative algebra, with an emphasis on labored examples and purposes. Their therapy combines based algebraic conception with functions to quantity conception, difficulties in classical Greek geometry, and the idea of finite fields, which has vital makes use of in different branches of technological know-how. issues lined comprise earrings and Euclidean earrings, the four-squares theorem, fields and box extensions, finite cyclic teams and finite fields. the cloth can serve both good as a textbook for a whole direction or as instruction for the extra learn of summary algebra.
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Extra info for An introductory course in commutative algebra
2 When the K-space Mn (K) is identiﬁed with K n , the polynomial ring K[xij | i, j = 1, 2, . . , n] can be used to introduce the Zariski topology on Mn (K). Here the matrix units eij form the standard basis of Mn (K) and they correspond to the variables xij . The closure of a subset A of Mn (K) is denoted by A. It is well known that right (and left) multiplication by any a ∈ Mn (K) is a continuous map in Zariski topology. More generally, if A ⊆ B are subsets of Mn (K), then A is closed in B if it is of the form X ∩ B for a closed subset X of Mn (K).
1. If K[G] is prime (semiprime) then K[S] is prime (semiprime, respectively). The converse holds provided G is the group of right quotients of S. 2. The group algebra is prime if and only if ∆(G) is torsion free. 3. The group algebra K[G] is semiprime if and only if char(K) = 0 or char(K) = p > 0 and ∆+ (G) is a p -group. 4. If G is the group of right quotients of S and K[G] is right Noetherian then B(K[S]) = B(K[G]) ∩ K[S] and B(K[S]) is nilpotent. 3. Group algebras of polycyclic-by-ﬁnite groups 45 Later we shall need an extension of the above theorem to a more general context.
Similarly one deﬁnes left Noetherian rings (respectively left Artinian rings). 1. Noetherian rings and rings satisfying a polynomial identity 39 and left Noetherian (respectively Artinian). A right Artinian ring is right Noetherian. 4]). 1. A right Noetherian ring contains only ﬁnitely many minimal prime ideals. 7]). The requirement for the existence of an identity is of course not assumed for nil subrings. 2. Nil subrings of right Noetherian rings are nilpotent. In particular the prime radical of a right Noetherian ring is nilpotent.