# Algebra II. Ring Theory: Ring Theory by Carl Faith

By Carl Faith

Ebook via religion, Carl

**Read or Download Algebra II. Ring Theory: Ring Theory PDF**

**Best algebra & trigonometry books**

**Spectral theory of automorphic functions**

Venkov A. B. Spectral thought of automorphic features (AMS, 1983)(ISBN 0821830783)

**Diskrete Mathematik fuer Einsteiger**

Dieses Buch eignet sich hervorragend zur selbstständigen Einarbeitung in die Diskrete Mathematik, aber auch als Begleitlektüre zu einer einführenden Vorlesung. Die Diskrete Mathematik ist ein junges Gebiet der Mathematik, das eine Brücke schlägt zwischen Grundlagenfragen und konkreten Anwendungen. Zu den Gebieten der Diskreten Mathematik gehören Codierungstheorie, Kryptographie, Graphentheorie und Netzwerke.

The 1st 3 chapters of this paintings comprise an exposition of the Wedderburn constitution theorems. bankruptcy IV comprises the speculation of the commutator subalgebra of an easy subalgebra of an ordinary uncomplicated algebra, the examine of automorphisms of an easy algebra, splitting fields, and the index relief issue idea.

**Extra info for Algebra II. Ring Theory: Ring Theory**

**Example text**

Then the following properties hold: (1) every left (resp. right) A-module M has a natural structure of a k-vector space, induced by scalar multiplication. Moreover, M is ﬁnite dimensional over k whenever it is ﬁnitely generated as an A-module; (2) A has a minimal left (resp. right) ideal; (3) for all n, m ≥ 1, we have An ∼ =A Am ⇐⇒ n = m. Proof. The ﬁrst point is obvious. To prove (2), observe that any left (resp. right) ideal of A is in particular a linear subspace of A, hence ﬁnite dimensional over k since A is.

Proof. 8, so D contains R. Notice also that Z(D) is a ﬁeld containing R, so Z(D) is isomorphic to either R or C, since Z(D)/R has ﬁnite degree by assumption. If D is commutative, then D is a ﬁeld extension of R, hence is isomorphic to R or C. Assume now that D is not commutative, and let k = Z(D). Then k cannot be isomorphic to C. Otherwise, for all d ∈ D, the minimal polynomial μd,k of d would be of degree 1, meaning that d ∈ k. We then would have that D = k, contradicting the fact that D is not commutative.

Since L is a simple k-subalgebra of A and ZA (L) = L (since L is commutative), the last part of the Centralizer Theorem gives ∼L Mr (L ⊗L CA (L)). A ⊗k L = Since L ⊗L CA (L) ∼ =L CA (L), we are done. Exercises 1. Let A be a central simple k-algebra. Show that the group of all k-algebra automorphisms of A is canonically isomorphic to A× /k× . 2. Let A be a simple k-algebra, not necessarily central. Is every automorphism of A inner? 3. The goal of this exercise is to provide an elementary and totally explicit proof of Skolem-Noether’s theorem for matrix algebras.