1.2 Motivations
2.1.1 Rings and Ideals
Definition 2.1.6 (Ring) A ring R is a set with two binary operations (addition, +, andmultiplication, ·) such that we have the following:
1. Ris an Abelian group with respect to addition. That is,Rhas a zero element 0, and everyx∈R has an additive inverse−x.
Section 2.1 Introduction to Rings and Ideals 27
2. R is a semigroup with respect to multiplication. Furthermore, mul- tiplication is distributive over addition:
∀x, y, z∈R
h
[x·(y+z) =x·y+x·z] and [(y+z)·x=y·x+z·x]i. We sayR has anidentity element if there is a 1∈Rsuch that
∀x∈R hx1 = 1x=xi.
The ring R is commutative if the multiplicative semigroup (R, ·) is com-
mutative:
∀x, y ∈R hxy=yxi.
The group (R, +, 0) is known as theadditive group of the ringR.
Some examples of rings are the following: the integers, Z, the rational numbers, Q, the real numbers, R, the complex numbers, C, polynomial functions innvariables over an ambient ringR,R[x1,. . .,xn] and rational functions innvariables over an ambient ringR,R(x1,. . .,xn). The set of
even numbers forms a ring without identity.
An interesting example of a finite ring,Z⋗, can be constructed by con- sidering the residue classes of Z mod m. The residue class containing i is
[i]m=i+mZ={i+⋗k:k∈Z}.
We can define addition and multiplication operations on the elements of
Z⋗ as follows:
[i]m+ [j]m= [i+j]m and [i]m·[j]m= [ij]m.
It can be easily verified that Z⋗, as constructed above, is a commutative ring with zero element [0]mand identity element [1]m; it is called thering of residue classes modm. Z⋗ is a finite ring withmelements: [0]m, [1]m, . . ., [m−1]m. For the sake of convenience,Z⋗ is often represented by the reduced system of residues modm, i.e., the set{0, 1,. . .,m−1}.
In what follows we assume that all of our rings are com- mutative and include an identity element. Any violation of this assumption will be stated explicitly.
Asubring R′ of a ringR is a nonempty subset ofR with the addition and multiplication operations inherited from R, which satisfies the ring postulates of Definition 2.1.6.
Definition 2.1.7 (Ideal) A subset I ⊆ R is an ideal if it satisfies the following two conditions:
1. I is an additive subgroup of the additive group ofR:
∀a, b∈I ha−b∈Ii.
2. RI ⊆I;I is closed under multiplication with ring elements:
∀a∈R ∀b∈I hab∈Ii.
The ideals{0} and R are called theimproper ideals of R; all other ideals areproper.
A subsetJ of an idealI in Ris asubideal ofI ifJ itself is an ideal in R. We make the following observations:
1. IfI is an ideal ofR, thenI is also asubring ofR.
2. The converse of (1) is not true; that is, not all subrings ofRare ideals. For example, the subringZ⊂Qis not an ideal of the rationals. (The set of integers is not closed under multiplication by a rational.) Let a ∈ R. Then the principal ideal generated by a, denoted (a), is given by
(a) ={ra:r∈R}, if 1∈R.
The principal ideal generated by zero element is (0) = {0}, and the prin- cipal ideal generated by identity element is (1) =R. Thus, the improper ideals of the ringR are (0) and (1).
Leta1,. . ., ak ∈R. Then theideal generated by a1, . . .,ak is (a1, . . . , ak) = nXk i=1 riai : ri∈R o .
A subsetF ⊆Ithat generatesIis called abasis(or, asystem of generators) of the idealI.
Definition 2.1.8 (Noetherian Ring) A ring R is called Noetherian if any ideal ofRhas a finite system of generators.
Definition 2.1.9
An elementx∈Ris called azero divisor if there existsy6= 0 inRsuch thatxy= 0.
An element x∈R isnilpotent ifxn = 0 for some n >0. A nilpotent element is a zero divisor, but not the converse.
An element x∈R is aunit if there existsy∈Rsuch thatxy= 1. The elementy is uniquely determined byxand is written asx−1. The units of Rform a multiplicative Abelian group.
Section 2.1 Introduction to Rings and Ideals 29
Definition 2.1.10
A ringR is called anintegral domain if it has no nonzero zero divisor. A ringR is calledreduced if it has no nonzero nilpotent element. A ringR is called afield if every nonzero element is a unit.
In an integral domainR,R\ {0}is closed under multiplication, and is denoted byR∗; (R∗,·) is itself a semigroup with respect to multiplication. In a field K, the group of nonzero elements, (K∗, ·, 1) is known as the multiplicative group of the field.
Some examples of fields are the following: the field of rational numbers,
Q, the field of real numbers,R, and the field of complex numbers,C. If p is a prime number, then Zp (the ring of residue classes mod p) is a finite field. If [s]p ∈Z∗
p, then the set of elements
[s]p, [2s]p, . . . , [(p−1)s]p
are all nonzero and distinct, and thus, for somes′∈[1..p−1], [s′s]p= [1]p; hence, ([s]p)−1= [s′]p.
Asubfieldof a field is a subring which itself is a field. IfK′is a subfield ofK, then we also sayK is anextension field ofK′. Leta∈K; then the smallest subfield (under inclusion) of K containingK′∪ {a} is called the extension of K′ obtained by adjoining atoK′, and denoted byK′(a).
The set of rationals,Q, is a subfield of the field of real numbers,R. If we adjoin an algebraic number, such as √2, to the field of rationals, Q, then we get an extension field,Q(√2)⊆R.
Definition 2.1.11 A field is said to be aprime field, if it does not contain any proper subfield. It can be shown that every fieldK contains a unique prime field, which is isomorphic to eitherQorZp, for some prime number p. We say the following:
1. A field K is ofcharacteristic 0 (denoted characteristicK = 0) if its prime field is isomorphic toQ.
2. A field K is of characteristic p >0 (denoted characteristicK =p), if its prime field is isomorphic to Zp.
Proposition 2.1.1 R6={0} is a field if and only if 1 ∈R and there are no proper ideals inR.
proof.
(⇒) LetR be a field, andI⊆R be an ideal ofR. Assume thatI 6= (0). Hence there exists a nonzero elementa∈I. Therefore, 1 =aa−1
∈I, i.e., I= (1) =R.
(⇐) Leta∈R be an arbitrary element ofR. Ifa6= 0, then the principal ideal (a) generated byamust be distinct from the improper ideal (0). Since R has no proper ideal, (a) =R. Hence there exists anx ∈R such that xa= 1, andahas an inverse inR. ThusR is a field.
Corollary 2.1.2 Every field K is a Noetherian ring.
proof.
The ideals ofK are simply (0) and (1), each of them generated by a single element.
Let R 6= {0} be a commutative ring with identity, 1 and S ⊆ R, a multiplicatively closed subset containing 1 (i.e., if s1 and s2 ∈ S, then s1·s2 ∈ S.) Let us consider the following equivalence relation “∼” on R×S: ∀ hr1, s1i,hr2, s2i ∈R×S h hr1, s1i ∼ hr2, s2i iff (∃s3∈S) [s3(s2r1−r2s1) = 0] i . LetRS =R×S/∼be the set of equivalence classes onR×Swith respect to the equivalence relation ∼. The equivalence class containing hr, si is denoted by r/s. The addition and multiplication on RS are defined as follows: r1 s1 +r2 s2 = s2r1+r2s1 s1s2 and r1 s1 · r2 s2 = r1r2 s1s2 .
The element 0/1 is the zero element ofRS and 1/1 is the identity element ofRS. It is easy to verify that RS is a commutative ring. The ringRS is called thering of fractions or quotient ring of Rwith denominator set S. IfSis chosen to be the multiplicatively closed set of all non-zero divisors of R, then RS is said to be the full ring of fractions or quotient ring of R, and is denoted by Q(R). In this case, the equivalence relation can be simplified as follows: ∀ hr1, s1i,hr2, s2i ∈R×S h hr1, s1i ∼ hr2, s2i iff s2r1=r2s1 i .
IfD is an integral domain andS=D∗, then DS can be shown to be a field;DS is said to be the field of fractions or quotient field of D, and is denoted byQF(D). The map
i : D→QF(D) : d7→d/1
defines an embedding of the integral domain D in the field QF(D); the elements of the form d1 are the “improper fractions” in the fieldQF(D).
For example, if we chooseD to be the integersZ, thenQF(Z) isQ, the field of rational numbers.
Section 2.1 Introduction to Rings and Ideals 31