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Unformatted text preview: Chapter 6 Rings
6.1. A Recollection of Rings
We encountered the deﬁnitions of rings and ﬁelds in Section 1.11. Let us
recall them here for convenience.
Deﬁnition 6.1.1. A ring is a nonempty set R with two operations: addition, denoted here by C, and multiplication, denoted by juxtaposition,
satisfying the following requirements:
(a) Under addition, R is an abelian group.
(b) Multiplication is associative.
(c) Multiplication distributes over addition: a.b C c/ D ab C ac ,
and .b C c/a D ba C ca for all a; b; c 2 R.
A ring is called commutative if multiplication is commutative, ab D
ba for all elements a; b in the ring. Recall that a multiplicative identity in
a ring is an element 1 such that 1a D a1 D a for all elements a in the ring.
An element a in a ring with multiplicative identity 1 is a unit or invertible
if there exists an element b such that ab D ba D 1.
Some authors include the the existence of a multiplicative identity in
the deﬁnition of a ring, but as this requirement excludes many natural examples, we will not follow this practice.
Let’s make a few elementary deductions from the ring axioms: Note
that the distributive law a.b Cc/ D ab Cac says that the map La W b 7! ab
is a group homomorphism of .R; C/ to itself. It follows that La .0/ D 0
and La . b / D La .b/ for any b 2 R. This translates to a 0 D 0 and
a. b / D ab . Similarly, Ra W b 7! ba is a group homomorphism of
.R; C/ to itself, and, consequently, 0 a D 0, and . b /a D b a. For n 2 Z
and a 2 R, since nb is the n–th power of b in the abelian group .R; C/,
we also have La .nb/ D nLa .b/; that is, a.nb/ D n.ab/. Similarly,
Ra .nb/ D nLa .b/; that is, .nb/a D n.ba/.
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