If R is a principal ideal domain and a, b are two non zero elements of R then show that [a, b](a, b) =abu where u is unit and (a, b) =g. c. d {a, b} , [a, b]=L. C. M{a,b}.
Answers
Answer:
Step-by-step explanation:
Chapter 1
Algebraic Systems
1.1 Operations, Algebraic Systems
An operation f on the set A is a map f : An → A where n is a natural number n ≥ 0. The number n
is called the arity of f.
n = 1 : f : A → A and f is just a map on A,
n = 2 : f : A2 → A and f is a binary operation on A.
The case n = 0 deserves special attention: For any set S one has that AS = {α|α : S → A}. In
particular, for n = {0, 1, . . . , n − 1},
A
n = {α|α : {0, 1, . . . , n − 1} → A} = {α = (a0, . . . , an−1)|aν ∈ A}
and this is the set of all n − tuples (a0, . . . , an−1) of elements in A.
An n−ary operation assigns to any n−tuple (a0, . . . , an−1) of elements in A an element f(a0, . . . , an−1
as operation value.
Now, A∅ = {α|α : ∅ → A} = {∅}. A nullary operation assigns to the empty map an element a ∈ A . A
nullary operation is therefore called a constant.
For A = ∅ one has that A∅ = {∅} and An = ∅ for n > 0. That is, only operations of positive arity
exist.
An algebraic system consists of
1. a set A,
2. a family (ft) of nt − ary operations on A.
We use the notation
A = (A,(ft)t∈T )
and ∆ = (nt)t∈T is called the similarity type of A.
Examples
1. Z = (Z,(f1, f2, f3)) where f1 : Z
2 → Z, (n, m) 7→ n + m, f2 : Z → Z, x 7→ −x, f3 : {∅} →
Z, ∅ 7→ 0. The binary operation f1 is the addition, the unary operation f2 is the additive inverse
and f3 is the zero.
We use for binary operations most of the time the symmetric notation xfy instead of f(x, y).