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Lecture 22
More on Rings
Last time:
A ring is a set R with two binary operations, and , such that
pR, q is an abelian group, pR, q is a semigroup (no identity or
inverses), and the distributive property holds.
Last time:
A ring is a set R with two binary operations, and , such that
pR, q is an abelian group, pR, q is a semigroup (no identity or
inverses), and the distributive property holds.
A zero divisor is an element a P R such that ab 0 or ba 0 for
some b. A unit is an element of a ring with 1 that has an inverse.
Last time:
A ring is a set R with two binary operations, and , such that
pR, q is an abelian group, pR, q is a semigroup (no identity or
inverses), and the distributive property holds.
A zero divisor is an element a P R such that ab 0 or ba 0 for
some b. A unit is an element of a ring with 1 that has an inverse.
A ring is commutative if pR, q is commutative. (e.g. Z)
An integral domain is where R has no zero divisors. (e.g. Z)
(Recall: I.D.’s have cancellation properties for non-zero-divisors.)
A division ring is where pR, q is a group. (e.g. Q)
A field is where pR, q is an abelian group. (e.g. Q)
More examples, more formality: Polynomial rings
Definition
Let R be a commutative ring with identity. The formal sum
an xn
with n ¥ 0 and each ai
coefficients ai in R.
an1 xn1
a1 x
a0
P R is called a polynomial in x with
More examples, more formality: Polynomial rings
Definition
Let R be a commutative ring with identity. The formal sum
an xn
an1 xn1
a1 x
a0
with n ¥ 0 and each ai P R is called a polynomial in x with
coefficients ai in R.
If an 0, the polynomial is of degree n, an xn is the leading term
and an is the leading coefficient. The polynomial is monic if
an 1.
More examples, more formality: Polynomial rings
Definition
Let R be a commutative ring with identity. The formal sum
an xn
an1 xn1
a1 x
a0
with n ¥ 0 and each ai P R is called a polynomial in x with
coefficients ai in R.
If an 0, the polynomial is of degree n, an xn is the leading term
and an is the leading coefficient. The polynomial is monic if
an 1.
The set of all such polynomials is the ring of polynomials in the x
with coefficients in R, denoted Rrxs. The ring R appears in Rrxs
as the constant polynomials.
More examples, more formality: Polynomial rings
If R is not an integral domain, then neither is Rrxs.
More examples, more formality: Polynomial rings
If R is not an integral domain, then neither is Rrxs.
If S
¤ R is a subring, then S rxs ¤ Rrxs is a subring.
More examples, more formality: Polynomial rings
If R is not an integral domain, then neither is Rrxs.
If S
¤ R is a subring, then S rxs ¤ Rrxs is a subring.
Proposition
Let R be an integral domain and let ppxq, q pxq P Rrxszt0u. Then
1. degree ppxqq pxq degree ppxq
degree q pxq,
2. the units of Rrxs are the units of R,
3. Rrxs is an integral domain.
More examples, more formality: Other rings with variables
Let R be a ring with 1.
The field of fractions of Rrxs is
tppxq{qpxq | ppxq, qpxq P Rrxs, qpxq 0u.
More examples, more formality: Other rings with variables
Let R be a ring with 1.
The field of fractions of Rrxs is
tppxq{qpxq | ppxq, qpxq P Rrxs, qpxq 0u.
The field of fractions over a field F is written F pxq.
The field of fractions of Zrxs is Qpxq.
More examples, more formality: Other rings with variables
Let R be a ring with 1.
The field of fractions of Rrxs is
tppxq{qpxq | ppxq, qpxq P Rrxs, qpxq 0u.
The field of fractions over a field F is written F pxq.
The field of fractions of Zrxs is Qpxq.
The ring of formal power series over R is
Rrrxss # 8̧
an x
n
| an P R
+
.
n 0
(“formal” means need not deal with convergence)
More examples, more formality: Other rings with variables
Let R be a ring with 1.
The field of fractions of Rrxs is
tppxq{qpxq | ppxq, qpxq P Rrxs, qpxq 0u.
The field of fractions over a field F is written F pxq.
The field of fractions of Zrxs is Qpxq.
The ring of formal power series over R is
Rrrxss # 8̧
an x
n
| an P R
+
.
n 0
(“formal” means need not deal with convergence)
If F is a field, the field of fractions over F rrxss is
F ppxqq #¸
¥
n N
an xn | an
+
P F, N P Z
.
More examples, more formality: Matrix rings
Definition
Let R be a ring and n a positive integer. Let Mn pRq be the set of
all n n matrices with entries from R.
More examples, more formality: Matrix rings
Definition
Let R be a ring and n a positive integer. Let Mn pRq be the set of
all n n matrices with entries from R.
Facts:
The set Mn pRq forms a ring under addition and multiplication.
It is not commutative, and it is not a division ring.
More examples, more formality: Matrix rings
Definition
Let R be a ring and n a positive integer. Let Mn pRq be the set of
all n n matrices with entries from R.
Facts:
The set Mn pRq forms a ring under addition and multiplication.
It is not commutative, and it is not a division ring.
The scalar matrices are those diagonal matrices of the form a id
with a P R. They form a subring of Mn pRq isomorphic to R (we’ll
define isomorphic later)
More examples, more formality: Matrix rings
Definition
Let R be a ring and n a positive integer. Let Mn pRq be the set of
all n n matrices with entries from R.
Facts:
The set Mn pRq forms a ring under addition and multiplication.
It is not commutative, and it is not a division ring.
The scalar matrices are those diagonal matrices of the form a id
with a P R. They form a subring of Mn pRq isomorphic to R (we’ll
define isomorphic later)
Other subrings: upper and lower triangular matrices, and if S
then Mn pS q.
¤ R,
More examples, more formality: Group rings
Fix a ring R with identity and a finite group G.
Let
#
RG ¸
P
ag g | ag
g G
be the group ring of G over R.
P R, g P G
+
More examples, more formality: Group rings
Fix a ring R with identity and a finite group G.
Let
#
RG ¸
P
ag g | ag
P R, g P G
g G
be the group ring of G over R.
With addition
component-wise:
°
°
if a gPG ag g and b gPG bg g, then
a
b
¸
P
g G
pa g
bg qg
+
More examples, more formality: Group rings
Fix a ring R with identity and a finite group G.
Let
#
RG ¸
P
ag g | ag
P R, g P G
g G
be the group ring of G over R.
With addition
component-wise:
°
°
if a gPG ag g and b gPG bg g, then
a
b
¸
P
pa g
bg qg
g G
and multiplication like polynomials:
pag gqpbbhq pag bhqpghq
and expansion according to distributive laws.
+