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Rings and Modules (6CCM350A/7CCM350B). Lecture 1, 24th September 2012
Lecturer: Alexander Paulin, room S413a. Email: [email protected]
Definition: A binary operation on a set G is a map ∗ : G×G → G. For g, h ∈ G we write g∗h := ∗(g, h).
Definition 1.1: (of a group): A group is a pair (G, ∗) where G is a set equipped with a binary operation
∗ such that
(i) The binary operation ∗ is associative on the set G, in other words
(∀x, y, z ∈ G) (x ∗ y) ∗ z = x ∗ (y ∗ z)
(ii) There exists an identity element e = 1G for the binary operation ∗, in other words
(∀g ∈ G) (g ∗ e = e ∗ g = g)
(iii) Every element g ∈ G has an inverse for the binary operation ∗, in other words
(∀g ∈ G) (∃g 0 ∈ G) (g ∗ g 0 = g 0 ∗ g = e)
An example of group is given by the pair (Z, +) where Z is the set of all integers, Z = {. . . , −2, −1, 0, 1, 2, 3, . . .},
and the binary operation is the standard addition of integers. The addition of integers is associative, the
identity element is 0 and the inverse of the integer n is −n.
Definition 1.2 (of a semi group): A semi group is a pair (G, ∗), as above, satisfying
(i) The binary operation ∗ is associative on the set G.
An example of semi group is the set of natural numbers N := {1, 2, 3, . . .} with the standard addition of
integers. We do not have an identity element and consequently the concept of inverses is meaningless. We
should note that being a semi-group does not mean that properties (ii) and (iii) are not satisfied. Any
group is automatically a semi-group.
Definition 1.3 (of monoid): A monoid is a pair (G, ∗), as above, satisfying
(i) The binary operation ∗ is associative on the set G.
(ii) There exists an identity element e = 1G for the binary operation ∗.
An example of monoid is the set of natural integers N = {0, 1, 2, . . .} together with the standard multiplication of integers. The neutral element in this case is 1.
A group (G, ∗) is called commutative or abelian if the binary operation ∗ is commutative, in other
words
(∀g, g 0 ) (g ∗ g 0 = g 0 ∗ g)
Definition 1.4 (of a ring): A ring is a triplet (R, +, ∗) such that
(i) (R, +) is a commutative group.
(ii) (R, ∗) is a semi group, in other words, the multiplication is associative.
(iii) The operation ∗ is left and right distributive over +, in other words,
r ∗ (s + t) = r ∗ s + r ∗ t
(s + t) ∗ r = s ∗ r + t ∗ r
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The fundamental example of a ring is (Z, +, ×), where + and × are the standard addition and multiplication of integers. This is why we use the + notation for general rings even though is may have no relation
to ordinary addition. With this in mind, if (R, +, ∗) is a ring, then we refer to + as addition and ∗ as
multiplication.
A more sophisticated example is the set of n×n matrices with complex entries, Mn (C). This has a natural
ring structure taking addition and multiplication to be ordinary addition and multiplication of matrices.
I leave it as an exercise to check all the axioms hold in this case.
A ring R is said to be a ring with unity if (R, ∗) is a monoid. In other words, that there is an
identity element for the multiplication. A ring R is said to be commutative if ∗ is commutative. (The
commutativity of a ring is a property of the multiplication, since by definition, the addition is always
commutative.) For example Mn (C) is not commutative are matrix multiplication is not. An example of
a ring without a multiplicative identity is 2Z By definition
2Z = {n ∈ Z : n is even}
Notation/Convention:
(i) We will sometimes write rs instead of r ∗ s.
(ii) If R is a ing with unity we will denote by 1 = 1R the identity element of (R, ∗), in other words the
identity element of the multiplication. We call 1 the multiplicative identity.
(iii) We will denote by 0 = 0R the identity element of (R, +), in other words the identity element of the
addition. We call 0 the additive identity.
(iv) By definition (R, +) is a commutative group. Hence every element r ∈ R has an inverse under
addition. We will denote by −r the inverse of r under addition. We call −r the additive inverse
of r.
(v) If R is a ring with unity, then (R, ∗) is merely a monoid. Hence not every element r ∈ R will have
an inverse under ∗. If R does have an inverse under multiplication, we say that r is a unit. In this
case we denote by r−1 , the multiplicative inverse of r. As an example the only units in (Z, +, ×)
are +1 and −1. We denote by R∗ the set of units in R.
It is very important to understand that even though we are using the same notation as for ordinary
addition and multiplication of the integers, in general they have nothing to do with each other. For
example, in a general ring, 1, 0 ∈ R have noting to do with 1, 0 ∈ Z. For example 1 ∈ Mn (C) is the
identity matrix.
Proposition: Let (R, +, ∗) be a ring with unity, then ∗ restricts to a binary operation on R∗ ⊂ R.
Moreover this makes (R∗ , ∗) a group.
Proof: Let r, s ∈ R∗ . Note that (rs)(s−1 r−1 ) = r(ss−1 )r−1 = rr−1 = 1 and (s−1 r−1 )rs = s−1 (r−1 r)s =
s−1 s = 1. Thus rs ∈ R∗ . Clearly 1 ∈ R∗ . Hence multiplication gives a binary operation on R∗ which is
assocaitive, has an identity, and by construction all elements have inverses.
Proposition: Let (R, +, ∗) be a ring, then
(i) For any r ∈ R, we have 0 ∗ r = r ∗ 0 = 0
(ii) For any r, s ∈ R, we have (−r) ∗ s = r ∗ (−s) = −(r ∗ s)
Proof:
(i) Given r ∈ R, we have 0r = (0 + 0)r = 0r + 0r. From 0r = 0r + 0r we extract 0r = 0.
(ii) Given r, s ∈ R, we have (−r)s + rs = (−r + r)s = 0s = 0. The first equality is true by distributivity
of the multiplication over the addition and the third equality is true because of the previous result.
From (−r)s + rs = 0, we have (−r)s = −(rs). Similarly we have r(−s) + rs = 0
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Consider a ring with unity R where 1 = 0. Then by tgiven r ∈ R we have r = 1 ∗ r = 0 ∗ r = 0. Thus
every element of R is equal to 0. This ring with one element is called the trivial ring. This also shows
that in a non-trivial ring 0 ∈
/ R∗ .
Definition 1.5 (of skew field and field): A non-trivial, ring with unity (R, +, ∗) is said to be a skew
field or a division ring if R − {0} = R∗ , i.e. every non-zero element has a multiplicative inverse. A
commutative skew field is simply called a field . Remember that commutative means ∗ is commutative.
The difference between skew fields and fields is that skew fields are not necessary commutative (but every
element has a multiplicative inverse in both cases).
Examples:
(i) Q, R, C are fields.
(ii) Z is a ring but not a field. For example 2 ∈ Z has no mulitplicative inverse in Z.
Definition 1.6 (of integral domain): Let R be a ring
(i) An element r is said to be a zero divisor sometimes written 0-divisor, if there exists a non zero
element s ∈ R, s 6= 0, such that either r ∗ s = 0 or s ∗ r = 0. Note that 0 itself is always a zero
divisor.
(ii) A commutative ring R is called integral domain or simply domain if 0 is the only zero divisor.
Proposition 1.7: A field is an integral domain.
Proof: Let R be a field. By definition R is commutative. We must show that the only zero divisors is
0. Recall that multiplication on R restricts to a binary operation on R∗ . Because R is a field we know
that R − {0} = R∗ . Thus multiplying any two non-zero elements must give a non-zero element. Hence
the only zero divisor is 0. Hence R is an integral domain.
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