Algebraic structures
Chapter 1: Rings
Definition, examples, elementary properties
I.1.1 A ring (with 1) (also called a unitary ring) is a five-tuple (R, +, ·, 0, 1) with R a set;
+ : R × R → R, (a, b) 7→ a + b; · : R × R → R, (a, b) 7→ ab; and 0, 1 ∈ R such that the
following properties hold:
1. (R, +, 0) is an Abelian group:
(a) a + (b + c) = (a + b) + c for all a, b, c ∈ R
(b) 0 + a = a + 0 = a for all a ∈ R
(c) Every a ∈ R has an ”opposite” −a ∈ R satisfying a + (−a) = (−a) + a = 0
(d) a + b = b + a for all a, b ∈ R
2. a(bc) = (ab)c for all a, b, c ∈ R
3. a(b + c) = (ab) + (ac) and (b + c)a = (ba) + (ca) for all a, b, c ∈ R
4. 1a = a1 = a for all a ∈ R
A ring is commutative if also
5. ab = ba for all a, b ∈ R
A division ring or skew field is a ring R such that in addition to 1, 2, 3, 4 as above, we
also have
6. 1 ̸= 0 and for all a ∈ R\{0}, there exists an inverse a−1 ∈ R satisfying aa−1 =
a−1 a = 1
A field is a commutative division field.
I.1.6 A subset R′ ⊂ R of a ring R is called a (unitary) subring of R if
1. 1 ∈ R′
2. R′ is a subgroup of the additive group of R, i.e. a + (−b) ∈ R′ for all a, b ∈ R′
3. ab ∈ R′ for all a, b ∈ R′
A subring R′ of a ring R is itself a ring, with the addition and multiplication of R. If R
is commutative, then so is R′ .
Z[i] = {a + bi | a, b ∈ Z} is called the ring of Gaussian integers.
Similarly, we have the sets
√ √ √ √
Z[ m] = {a + b m | a, b ∈ Z}, Q[ m] = {a + b m | a, b ∈ Q}
1
,I.1.9 Let R be a ring. For a, b, b1 , . . . , bn , c ∈ R we have
a(b1 + b2 + · · · + bn ) = ab1 + ab2 + · · · + abn
(b1 + b2 + · · · + bn )a = b1 a + b2 a + · · · + bn a
a(b − c) = ab − ac
a·0=0·a=0
Units and zero divisors
I.2.1 Let R be a unitary ting. An element a ∈ R is called a unit (or invertible) if some
b ∈ R exists such that ab = ba = 1. The set of units in R is called the unit group of R
and denoted R× or U (R).
An element a ∈ R is called a left unit if there exists b ∈ R such that ab = 1, and a right
unit if there exists b ∈ R such that ba = 1. If a ∈ R is both a left and right unit, then a
is a unit.
In general, R is a division ring if and only if R× = R\{0}.
I.2.3 The unit group R× of a ring R with 1 is a group with respect to multiplication.
If R is commutative, then R× is Abelian. The converse is not true.
I.2.6
• An element a ∈ R, R a ring, is called a left zero divisor if a ̸= 0 and there exists
b ∈ R, b ̸= 0 and ab = 0.
• Similarly, the element a is called a right zero divisor if a ̸= 0 and there exists a
non-zero x such that ca = 0.
• a is called a zero divisor if it is either a left or a right divisor (or both).
• A nilpotent element is an a ∈ R, a non-zero, such that an = 0 for some n ∈ N. In
particular, a nilpotent element is a zero divisor, both left and right.
• An element a ∈ R is called an idempotent element if a2 = a and 0 ̸= a ̸= 1. An
idempotent element is in particular a zero divisor (both left and right).
I.2.8 An element a ∈ R where R is a commutative ring with 1 cannot be both a zero
divisor and a unit. In a non-commutative ring, a left zero divisor is not a right unit and
a right zero divisor is not a left unit.
I.2.10 A division ring contains no zero divisors.
I.2.11 For n ∈ Z>0 one has that Z/nZ is a field if and only if n is a prime number. Given
a prime number p, one denotes the field Z/pZ as Fp and it consists of p elements.
I.2.13 An integral domain (or domain or integral ring) is a commutative ring with 1 ̸= 0
having no zero divisors.
2
, I.2.15 Suppose R is a ring without zero divisors and let a, b, c ∈ R. Then
• ab = 0 if and only if a = 0 or b = 0;
• ab = ac if and only if a = 0 or b = c.
Construction of rings
Products of rings
If R1 and R2 are rings and R = R1 × R2 , one defines multiplication and addition on R
as follows:
(r1 , r2 ) + (s1 , s2 ) = (r1 + s1 , r2 + s2 )
(r1 , r2 ) · (s1 , s2 ) = (r1 s1 , r2 s2 )
We have that 0 = (0, 0) and 1 = (1, 1). R is commutative if and only if both R1 and
R2 are commutative. Moreover, R× = R1× × R2× . A ring R1 × R2 with R1 ̸= {0} ̸= R2
contains zero divisors, and the elements (0, 1) and (1, 0) are idempotent.
Fields of fractions
Let R be a domain. Q(R) is called the field of fractions or quotient field, and it has the
following properties:
• R ⊂ Q(R)
• Every s ∈ R with s ̸= 0 has an inverse s−1 ∈ Q(R).
• Every element of Q(R) can be written as r · s−1 for some r, s ∈ R, s ̸= 0.
The construction generalises the construction of Q = Q(Z).
Let S = R\{0}. On R×S, define the equivalence relation ∼ by (a, s) ∼ b(t) ⇐⇒ at = bs.
Now let Q(R) denote the set of equivalence classes of ∼:
Q(R) = (R × S)/ ∼
For the equivalence class containing (a, s) we introduce the notation as , such that
na o a a b
Q(R) = : a, s ∈ R, s ̸= 0 , = {(b, t) ∈ R×S : (a, s) ∼ (b, t)}, = ⇐⇒ at = bs
s s s t
We define addition and multiplication on Q(R) by
a b at + bs a b ab
+ = , · =
s t st s t st
a
Then Q(R) is a field. R is a subring of Q(R) by identifying a ∈ R ⇐⇒ 1
∈ Q(R). For
s ∈ S, we have s−1 = 1s .
Any field K gives rise to a ring of polynomials with coefficients in K in the variable X,
called K[X]. The ring K[X] is a domain, and we define the field of rational functions in
one variable over K by K(X) := Q(K[X]).
3
Chapter 1: Rings
Definition, examples, elementary properties
I.1.1 A ring (with 1) (also called a unitary ring) is a five-tuple (R, +, ·, 0, 1) with R a set;
+ : R × R → R, (a, b) 7→ a + b; · : R × R → R, (a, b) 7→ ab; and 0, 1 ∈ R such that the
following properties hold:
1. (R, +, 0) is an Abelian group:
(a) a + (b + c) = (a + b) + c for all a, b, c ∈ R
(b) 0 + a = a + 0 = a for all a ∈ R
(c) Every a ∈ R has an ”opposite” −a ∈ R satisfying a + (−a) = (−a) + a = 0
(d) a + b = b + a for all a, b ∈ R
2. a(bc) = (ab)c for all a, b, c ∈ R
3. a(b + c) = (ab) + (ac) and (b + c)a = (ba) + (ca) for all a, b, c ∈ R
4. 1a = a1 = a for all a ∈ R
A ring is commutative if also
5. ab = ba for all a, b ∈ R
A division ring or skew field is a ring R such that in addition to 1, 2, 3, 4 as above, we
also have
6. 1 ̸= 0 and for all a ∈ R\{0}, there exists an inverse a−1 ∈ R satisfying aa−1 =
a−1 a = 1
A field is a commutative division field.
I.1.6 A subset R′ ⊂ R of a ring R is called a (unitary) subring of R if
1. 1 ∈ R′
2. R′ is a subgroup of the additive group of R, i.e. a + (−b) ∈ R′ for all a, b ∈ R′
3. ab ∈ R′ for all a, b ∈ R′
A subring R′ of a ring R is itself a ring, with the addition and multiplication of R. If R
is commutative, then so is R′ .
Z[i] = {a + bi | a, b ∈ Z} is called the ring of Gaussian integers.
Similarly, we have the sets
√ √ √ √
Z[ m] = {a + b m | a, b ∈ Z}, Q[ m] = {a + b m | a, b ∈ Q}
1
,I.1.9 Let R be a ring. For a, b, b1 , . . . , bn , c ∈ R we have
a(b1 + b2 + · · · + bn ) = ab1 + ab2 + · · · + abn
(b1 + b2 + · · · + bn )a = b1 a + b2 a + · · · + bn a
a(b − c) = ab − ac
a·0=0·a=0
Units and zero divisors
I.2.1 Let R be a unitary ting. An element a ∈ R is called a unit (or invertible) if some
b ∈ R exists such that ab = ba = 1. The set of units in R is called the unit group of R
and denoted R× or U (R).
An element a ∈ R is called a left unit if there exists b ∈ R such that ab = 1, and a right
unit if there exists b ∈ R such that ba = 1. If a ∈ R is both a left and right unit, then a
is a unit.
In general, R is a division ring if and only if R× = R\{0}.
I.2.3 The unit group R× of a ring R with 1 is a group with respect to multiplication.
If R is commutative, then R× is Abelian. The converse is not true.
I.2.6
• An element a ∈ R, R a ring, is called a left zero divisor if a ̸= 0 and there exists
b ∈ R, b ̸= 0 and ab = 0.
• Similarly, the element a is called a right zero divisor if a ̸= 0 and there exists a
non-zero x such that ca = 0.
• a is called a zero divisor if it is either a left or a right divisor (or both).
• A nilpotent element is an a ∈ R, a non-zero, such that an = 0 for some n ∈ N. In
particular, a nilpotent element is a zero divisor, both left and right.
• An element a ∈ R is called an idempotent element if a2 = a and 0 ̸= a ̸= 1. An
idempotent element is in particular a zero divisor (both left and right).
I.2.8 An element a ∈ R where R is a commutative ring with 1 cannot be both a zero
divisor and a unit. In a non-commutative ring, a left zero divisor is not a right unit and
a right zero divisor is not a left unit.
I.2.10 A division ring contains no zero divisors.
I.2.11 For n ∈ Z>0 one has that Z/nZ is a field if and only if n is a prime number. Given
a prime number p, one denotes the field Z/pZ as Fp and it consists of p elements.
I.2.13 An integral domain (or domain or integral ring) is a commutative ring with 1 ̸= 0
having no zero divisors.
2
, I.2.15 Suppose R is a ring without zero divisors and let a, b, c ∈ R. Then
• ab = 0 if and only if a = 0 or b = 0;
• ab = ac if and only if a = 0 or b = c.
Construction of rings
Products of rings
If R1 and R2 are rings and R = R1 × R2 , one defines multiplication and addition on R
as follows:
(r1 , r2 ) + (s1 , s2 ) = (r1 + s1 , r2 + s2 )
(r1 , r2 ) · (s1 , s2 ) = (r1 s1 , r2 s2 )
We have that 0 = (0, 0) and 1 = (1, 1). R is commutative if and only if both R1 and
R2 are commutative. Moreover, R× = R1× × R2× . A ring R1 × R2 with R1 ̸= {0} ̸= R2
contains zero divisors, and the elements (0, 1) and (1, 0) are idempotent.
Fields of fractions
Let R be a domain. Q(R) is called the field of fractions or quotient field, and it has the
following properties:
• R ⊂ Q(R)
• Every s ∈ R with s ̸= 0 has an inverse s−1 ∈ Q(R).
• Every element of Q(R) can be written as r · s−1 for some r, s ∈ R, s ̸= 0.
The construction generalises the construction of Q = Q(Z).
Let S = R\{0}. On R×S, define the equivalence relation ∼ by (a, s) ∼ b(t) ⇐⇒ at = bs.
Now let Q(R) denote the set of equivalence classes of ∼:
Q(R) = (R × S)/ ∼
For the equivalence class containing (a, s) we introduce the notation as , such that
na o a a b
Q(R) = : a, s ∈ R, s ̸= 0 , = {(b, t) ∈ R×S : (a, s) ∼ (b, t)}, = ⇐⇒ at = bs
s s s t
We define addition and multiplication on Q(R) by
a b at + bs a b ab
+ = , · =
s t st s t st
a
Then Q(R) is a field. R is a subring of Q(R) by identifying a ∈ R ⇐⇒ 1
∈ Q(R). For
s ∈ S, we have s−1 = 1s .
Any field K gives rise to a ring of polynomials with coefficients in K in the variable X,
called K[X]. The ring K[X] is a domain, and we define the field of rational functions in
one variable over K by K(X) := Q(K[X]).
3
