Abstract Algebra is a vast branch of mathematics which we can barely do justice to in one blog post. Also known as "modern algebra" it focuses on algebraic structures rather than specific number systems. Among these structures are rings, fields and groups - the first two of which will be the focus of this blog post. (Groups to follow.) For those with the ambition to delve into a full (.pdf) text book, you can go to this link:
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 - 1 Î S such that: a · a - 1 = 1. The element a - 1 is called the reciprocal or multiplicative inverse of a.
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) = a0 + a1 x + · · · + an xn 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 r1, r2, . . . , rn Î F such that
(i) f(x) = an (x-r1) (x-r2) · · · (x-rn), and
(ii) K = F (r1,r2,...,rn).
In the above situation we usually say that f(x) splits over the field F. The elements r1, r2, . . . , rn 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}.
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