Download A monologue - take 2? The study of Group and Ring theory is

A monologue - take 2?
The study of Group and Ring theory is essential to Abstract Algebra. Groups
and Rings are powerful tools for understanding how to transform a situation
given various degrees of freedom. For example, conservation laws of physics
are impressions of the concept of least action. Ideas in Ring theory are also
fundamental in number theory, projective geometry, topology and many other
fields. Hence, it’s worthwhile to try to understand how they work.
Using elementary principles in Group and Ring theory, we attempt to justify
the following claim. In Z,
0 = −0
Due to time constraints, some structural properties of Groups and Rings are
assumed and therefore ommitted, or not explored in depth
Prelim
A binary operation is an arity two operation whose two domains and codomain
are subsets of the same set.
Groups
A group G is a set, with a binary operation ∗. More formally,
Defn: Given a set G, a group is an ordered pair (G, ∗) with the following properties:
1. G is closed under ∗
2. ∗ is associative (on G)
3. Identity: ∃e ∈ G, such that for a ∈ G, a ∗ e = e ∗ a = a
4. Inverse: a ∈ G =⇒ ∃a−1 ∈ G
Uniqueness of the identity and inverse sometimes appears in alternate definitions of a group. However, that is unecessary, since they follow directly from
the above definition.
Claim: The identity is unique.
Pf: Let (G, ∗) be a group. e, e′ ∈ G. Suppose e and e′ are distinct identites for
G, hence e ̸= e′ .
e = e′ ∗ e (axiom 3, because e′ is an identity)
e′ ∗ e = e′ (axiom 3, because e is an identity)
e = e′ . Contradiction, in the hypothesis we supposed e ̸= e′ . Hence, the identity
is unique.
Using similar techniques, it’s possible to show the inverse is also unique.
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Defn: (G, ∗) is abelian/commutative if ∗ is commutative.
One of the most important commutative groups is Zm , the multiplicative
group of integers modulo m. This group appears heavily in number theory due
to it’s applications in crypography. In particular, when m is prime, Zp forms a
field....fuck this, i’m a girl, what do i know?
A fundamental non-commutative group to understand is the dihedral group.
Defined as Dn =< x, y|x2 = 1, y n = 1, (x, y)2 = 1 >, this group is the symmetry group of an n-sided regular polygon. The elements of the group are rotations
and reflections of the n-sided polygon.
Rings
A ring R is a set with two binary operations, ∗, +.
Defn: Given a set R, (R, +, ∗) is a ring if and only if
1. (R, +) is an abelian group.
2. ∗ is associative (on R)
3. distributive law holds for R:
(a + b) ∗ c = (a ∗ c) + (b ∗ c) and a ∗ (b + c) = (a ∗ b) + (a ∗ c)
Claim: (Z, +, ∗), where + and ∗ are the canonical arithmetic definitions of
addition and multiplication forms a ring.
Pf: ommited
The identity element in abelian group (Z, +) is denoted 0. For a ∈ G, the
additive inverse of a is −a. As such, −0 is the additive inverse of 0 in Z.
Confusion often arises due to abuse of notation. The additive inverse in a ring
can be calculated by multiplying by −1. In particular, −a = −1 ∗ a. It is also
very closely related to subtraction. This relationship can be outlined in the
following ways:
Claim: R ring. a − b = a + (−b)
Pf: excersize, can be shown from ring axioms and that −a = −1 ∗ a
Claim: −a = 0 − a
Pf: also excersize.
We are now prepared to answer why 0 = −0.
Claim: In (Z, +,∗), 0 = −0.
Pf: 0 = 0 + (−0)
−0 = 0 + (−0) (axiom 2 from definition of group)
Hence, 0 = −0.
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