*We now return to looking again at the mathematical entities known as rings, fields and ideals.*

*1.***:**

*Definitions*

*i)*

*Binary operation:*
Let S be an abstract set. By binary operation on S,

*denoted by**o*one means a mapping on a function from S x S to S. In other words, a binary operation assigns to each pair of elements in S: (a,b) an element a o b in S.

*ii)*

*Ring*

By a ring (S, +,

**· )**one means an abstract set**S**which has two binary operations defined on it. The operation + is called addition, and the operation**·**is called**multiplication.**Moreover, the set (S, +, ·) is required to satisfy the following axioms:
Let
a, b and c Î S

A1): a + (b + c) = (a + b) + c

A2) ∃ an
element 0 Î S with the property that 0 + a = a

A3) For each element a Î S there exists an element

(-a) Î S such that a + (-a) = 0

A4) a + b = b + a

A5) a · (b · c) = (a · b)
· c

*iii)*

*Commutative Ring*
By a
commutative ring (S, +

**· )**one means a ring (S, +**· )**which satisfies, in addition to the ring axioms the axiom:
A6) For
all a, b
Î S then a · b
= b · a

And there
also exist the properties:

*a)Closure:*If a,b Î S, then the sum a+b and the product a

**·**b are uniquely defined and belong to S.

*b)Associative laws:*For all a,b,c Î S,

a+(b+c) = (a+b)+c and a

**·**(b**·**c) = (a**·**b)**·**c.
c)

*Commutative laws:*For all a,b Î S, a+b = b+a and a**·**b = b**·**a.
d)

*Distributive laws*: For all a,b,c Î S,
a

**·**(b+c) = a**·**b + a**·**c and (a+b)**·**c = a**·**c + b**·**c
e)

*Additive identity*: The set R contains an**additive identity element**, denoted by 0, such that for all a Î S,
a+0
= a and 0+a = a.

f)

*Additive inverses*: For each a Î S, the equations:
a+x
= 0 and x+a = 0

have a solution x Î S, called the

**additive inverse**of a, and denoted by (-a).The commutative ring S is called a

**commutative ring with identity**if it contains an element 1, assumed to be different from 0, such that for all a Î S,

a

**·**1 = a and 1**·**a = a.
In this case, 1 is called a

**multiplicative identity element**

*iv)Commutative Ring with unity*
By a
commutative ring with unity one means a commutative ring (S, +

**· )**which satisfies the axiom:
A7) There
exists an element 1 Î S with the property:

1

**·**a = a for all a Î S

*v)Integral domain*
By an
integral domain ring (S, +

**· )**one means a commutative ring with unity which satisfies the following axiom:
If a, b Î S and a

**·**b = 0 then either a = 0 or b = 0.

*vi)Fields*
By a
field (S, +

**·**)**one means a****commutative ring**with unity which satisfies the additional axiom:
A7) For
every non-zero element a Î S there exists an
element a

**Î S such that: a · a**^{- 1}**= 1. The element a**^{- 1}**is called the reciprocal or multiplicative inverse of a.**^{- 1}**vii)**

*Ordered Fields*
An
ordered field is a field (S, +

**·**)**which contains a subset P (called the positive cone in P) which has the following properties:**
P is a
proper subset of S.

O.1) P is
closed under addition.

O.2) P is closed under multiplication

O.3)
If a
Î S then one and only one of the following three
conditions must hold:

a =
0, a
Î P, or - a Î P

*viii)Order relation on an ordered field*
Let (F , +

**·**)**be an ordered field and let P represent the positive cone in F. We define a relation ‘<’ on F, called ‘***less than’*by the formula:
< =
{(a, b) : a, b Î F, b – a Î P }

Or,
equivalently:

a < b if an only if b – a Î P

We also
define the relation ‘greater than’ on F by

> as follows:

s > b
if and only if b < a

**Definition.**The field K is said to be an

**extension field**of the field (F +

**·**)

**if (F +**

**·**)

**is a subset of K which is a field under the operations of K.**

**Definition.**Let K be an extension field of F and let u Î K. If there exists a nonzero polynomial f(x) Î F[x] such that f(u)=0, then u is said to be

**algebraic**over F. If there does not exist such a polynomial, then u is said to be

**transcendental**over K.

**Proposition.**Let K be an extension field of F , and let u Î K be algebraic over F. Then there exists a unique irreducible polynomial p(x) Î F[x] such that p(u)=0. It is characterized as the monic polynomial of minimal degree that has u as a root.

Furthermore, if f(x) is any polynomial in F[x] with f(u)=0, then p(x) | f(x).

**Definition.**Let F be a field and let f(x) = a

_{0}+ a

_{1}x + · · · + a

_{n }x

^{n}be a polynomial in F [x] of degree n> 0. An extension field K of F is called a

**splitting field for f(x) over F**if there exist elements r

_{1}, r

_{2}, . . . , r

_{n}Î F such that

**(i)**f(x) = a

_{n}(x-r

_{1}) (x-r

_{2})

**· · ·**(x-r

_{n}), and

**(ii)**K = F (r

_{1},r

_{2},...,r

_{n}).

In the above situation we usually say that f(x)

**splits**over the field F. The elements r_{1}, r_{2}, . . . , r_{n}are roots of f(x), and so K is obtained by adjoining to F a complete set of roots of f(x).**Definition: Subfield of a field.**Let (F +

**·**)

**be a field. A subset T ⊂ F is called a subfield if T is closed under the operations of + and**

**·**, and T is a field under those operations.

**Definition: Ideal:**Let (S, +

**·**)

**be a commutative ring, by an ideal I in S one means the following:**

i)I is a subring of S,

ii)If a Î
I , b Î S then a, b Î
I

**Definition: Congruence modulo I.**

Let S be a
commutative ring. Let I be an ideal in S. We write a

**≡**b modulo I for the two elements a, b Î S if
a Î [ b ]. Hence,
b Î [ a ].

**Definition:**S is a commutative ring and I is an ideal. Then let S/ I denote

**a quotient ring**and the set of congruence classes modulo I.

Perhaps the most
critical aspect of ideals to note is that:

*“an ideal is to a ring as a normal subgroup is to a group”*. Also, in a similar manner to groups, one can have left and right ideals.
Let a Î S,
a commutative ring. Then the set (a) defined by: (a) = {ax: , x Î S }
is an ideal.

Now, let S be a commutative ring and I be an ideal in S. For
each element a Î S let [a]
be the subset of S defined by:

[a] = I + a = {a + j:
j Î S}

Further, let s be a subring of a ring S satisfying:

[a] s ⊆
s
and s[a] ⊆ s for
all a
Î S is an ideal (or

*two-sided ideal*of S). Also, a subring s of S satisfying
[a] s ⊆
s
for all a Î
S is a

**left ideal**of S. One satisfying s [a] ⊆ s for all a Î S is a**right ideal**of S.

__Problems for Math Mavens____:__

1) Let Z be the
integers. Then prove for each class:

[a]
mod n there exists an integer r Î
Z, for: 0

__<__r < n, r Î [n]
2) Let Z be the
integers. The ideal:

(5) = {5
j: j Î Z }

Show

*all the congruence classes*with respect to this ideal.
Hint: [a] =
{a + j: j Î [(5)} =
{a + 5j:

**j Î Z }**
3) Let S be a commutative ring, and let I be an
ideal in S. If A, B are subset of S, then use set notation to define: i) A

**·**B, A + B
4) Take
S as the set of integers, Z. Let the ideal I = (2) so that S

**/**I = Z_{2}Thence or otherwise, find:
a) [0] b)
[1] c) S/ I =
Z

_{5}
5)
Show every ring S
has

*two ideals*: S itself and {0}.
## No comments:

Post a Comment