Mathematics Exam  >  Mathematics Notes  >  Algebra  >  Rings, Ideals, Quotient Rings - Ring Theory

Rings, Ideals, Quotient Rings - Ring Theory | Algebra - Mathematics PDF Download

A ring A is a set with + , • such that

(1) (A, +) is an abelian group;
(2) (A, •) is a semigroup;
(3) • distributes over + on both sides.
(4) (∀x, y ∈ A) x • y = y • x
(5) (∃1 ∈ A)(∀x ∈ A) 1•x = x•1 = x .


In this course all rings A are commutative, that is, and have an identity element 1 (easily seen to be unique)

If 1 = 0 then A = {0} (easy to see), called the zero ring.


Multiplication will be denoted by juxtaposition, and simple facts used without comment, such as

(∀x, y ∈ A)
x 0 = 0 ,
(−x)y = x(−y) = −(xy) ,
(−x)(−y) = xy 


Call a subset S of a ring A a subring if

(i) 1 ∈ S ;
(ii) (∀x, y ∈ S ) x + y , xy , −x ∈ S .


Condition (ii) is easily seen to be equivalent to

(ii)′ (∀x, y ∈ S ) x − y , xy ∈ S .

 

Note: In other contexts authors replace the condition 1 ∈ S by S ≠ ∅ (which is not equivalent!).


Examples:

(1) Z is the only subring of Z .

(2) Z is a subring of Q , which is a subring of R , which is a subring of C .

(3) Z[i] = { a + bi | a, b ∈ Z } Rings, Ideals, Quotient Rings - Ring Theory | Algebra - Mathematics

the ring of Gaussian integers is a subring of C 

(4) Zn = { 0, 1, . . . , n − 1}

with addition and multiplication mod n .

(Alternatively Zn may be defined to be the quotient ring Z/nZ , defined below)

(5) R any ring, x an indeterminate. Put

R[[x]] = {a0 + a1x + a2x2 + . . . | a0, a1, . . . ∈ R} , the set of formal power series over R , which becomes a ring under addition and multiplication of power series. Important subring:

R[x] = {a0 + a1x + . . . +anxn | n ≥ 0 , a0, a1, . . . , an ∈ R} 

the ring of polynomials over R .

Call a mapping f : A → B (where A and B are rings) a ring homomorphism if

(a) f (1) = 1 ;

(b) (∀x, y ∈ A) f (x + y) = f (x) + f (y)
and
f (xy) = f (x)f (y) ,


in which case the following are easily checked:

(i) f (0) = 0 ;
(ii) (∀x ∈ A) f (−x) = −f (x) ;
(iii) f (A) = {f (x) | x ∈ A} , the image of f is a subring of B ;
(iv) Composites of ring hom’s are ring hom’s.


An isomorphism is a bijective homomorphism, say f : A → B , in which case we write

A ≌ B or f : A ≌ B.

It is easy to check that

≌ is an equivalence relation.

A nonempty subset I of a ring A is called an ideal, written Rings, Ideals, Quotient Rings - Ring Theory | Algebra - Mathematics

(i) (∀x, y ∈ I ) x + y , −x ∈ I 

[clearly equivalent to (i)′ (∀x, y ∈ I ) x − y ∈ I ]:

(ii) (∀x ∈ I )(∀y ∈ A) xy ∈ I .


In particular I is an additive subgroup of A , so we can form the quotient group the group of cosets of I ,

A/I = { I + a | a ∈ A} ,

with addition defined by, for a, b ∈ A , (I + a) + (I + b) = I + (a + b) .

Further A/I forms a ring by defining, for a, b ∈ A ,

(I + a) (I + b) = I + (a b) .

Verification of the ring axioms is straightforward:


— only tricky bit is first checking multiplication is well-defined:

If I + a = I + a′ and I + b = I + b′ then

a − a′ , b − b′ ∈ I ,

so

ab − a′b′ = ab − ab′ + ab′ − a′b′

= a(b − b′) + (a − a′)b′ ∈ I 

yielding I + ab = I + a′b′ .

We call A/I a quotient ring.

The mapping

φ : A → A/I , x → I + x

is clearly a surjective ring homomorphism, called the natural map, whose kernel is

ker φ = {x ∈ A | I + x = I} = I .

Thus all ideals are kernels of ring homomorphisms.

The converse is easy to check, so

kernels of ring homomorphisms with domain A are precisely ideals of A .


The following important result is easy to verify:

Fundamental Homomorphism

Theorem: If f : A → B is a ring homomorphism with kernel I and image C then

A/I ≌ C .

 

Proposition: Rings, Ideals, Quotient Rings - Ring Theory | Algebra - Mathematics and φ : A → A/I be the natural map. Then(i) ideals J of A/I have the form J = J/I = {I + j | j ∈ J} for some J such that I ⊆ J  A ; (ii) φ−1 is an inclusion-preserving bijection between ideals of A/I and ideals of A containing I .

 

Rings, Ideals, Quotient Rings - Ring Theory | Algebra - Mathematics

Example: The ring Zn = {0, 1, . . . , n − 1} with mod n arithmetic is isomorphic to Z/nZ : follows from the Fundamental Homomorphism

Theorem, by observing that the mapping f : Z → Zn where

f (z) = remainder after dividing z by n is a ring homomorphism with image Zn and kernel nZ.

The document Rings, Ideals, Quotient Rings - Ring Theory | Algebra - Mathematics is a part of the Mathematics Course Algebra.
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FAQs on Rings, Ideals, Quotient Rings - Ring Theory - Algebra - Mathematics

1. What is a ring in mathematics?
Answer: In mathematics, a ring is a set equipped with two binary operations, addition and multiplication, that satisfy certain properties. These properties include closure under addition and multiplication, associativity, distributivity, and the existence of an additive identity and additive inverses. Rings are an important algebraic structure studied in abstract algebra.
2. What are ideals in ring theory?
Answer: In ring theory, ideals are subsets of a ring that have special properties. An ideal is a subset of a ring such that it is closed under addition, subtraction, and multiplication by any element of the ring. Additionally, it must contain the additive identity element of the ring. Ideals play a crucial role in the study of rings as they allow for the concept of quotient rings.
3. What is a quotient ring in ring theory?
Answer: A quotient ring, also known as a factor ring, is a ring that is obtained by "modding out" a ring by an ideal. Given a ring R and an ideal I, the quotient ring R/I consists of the cosets of I in R with addition and multiplication defined in a specific way. Quotient rings provide a way to partition a ring into smaller, more manageable pieces, and they retain the algebraic structure of the original ring.
4. How are ideals related to quotient rings?
Answer: Ideals are closely related to quotient rings. In fact, the set of cosets of an ideal I in a ring R forms a quotient ring, denoted by R/I. The elements of the quotient ring R/I are the cosets of I, and addition and multiplication are defined by operations on these cosets. The quotient ring allows us to study the structure and properties of a ring by considering its cosets and their algebraic operations.
5. What are some applications of ring theory in mathematics?
Answer: Ring theory has numerous applications in various branches of mathematics. It is used in algebraic geometry to study geometric objects defined by polynomial equations. Ring theory is also fundamental in number theory, where rings of integers, such as the ring of Gaussian integers, are studied. Additionally, ring theory plays a crucial role in algebraic coding theory, cryptography, and the study of algebraic structures in physics.
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