By A.I. Kostrikin, I.R. Shafarevich, E. Behr, Yu.A. Bakhturin, L.A. Bokhut, V.K. Kharchenko, I.V. L'vov, A.Yu. Ol'shanskij

Algebra II is a two-part survey almost about non-commutative jewelry and algebras, with the second one half inquisitive about the idea of identities of those and different algebraic structures. It presents a vast assessment of the main glossy developments encountered in non-commutative algebra, in addition to the varied connections among algebraic theories and different components of arithmetic. a big variety of examples of non-commutative earrings is given initially. in the course of the e-book, the authors comprise the historic historical past of the traits they're discussing. The authors, who're one of the such a lot widespread Soviet algebraists, proportion with their readers their wisdom of the topic, giving them a different chance to benefit the cloth from mathematicians who've made significant contributions to it. this can be very true on the subject of the idea of identities in sorts of algebraic items the place Soviet mathematicians were a relocating strength in the back of this method. This monograph on associative earrings and algebras, crew concept and algebraic geometry is meant for researchers and scholars.

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**Additional resources for Algebra II: Noncommutative Rings. Identities **

**Example text**

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.