The following glossary includes terminology covered thus far in class this semester. I have indicated
page numbers in the textbook where these notions are introduced; also page numbers in the handouts
distributed in class (and available on the course website). The relevant handouts are
A.
B.
C.
D.
E.
F.
G.
H.
Three Examples of Proofs
Mathematical Induction
Review: Basic Notation and Properties of Integers
Factorization in Rings
More Factorization in Rings
Basic Terminology and Results for Rings
Polynomials, Power Series and Such
Homomorphisms, Ideals and Quotients
GLOSSARY
Term / Concept
mathematical proof
mathematical induction
well ordering
Fundamental Theorem of Arithmetic
Euclid’s Lemma
Euclid’s Algorithm
handout / pages
A
B
B9, G9
B10, G8
G7
5
4
24, 58
23
18, 20
, , 0, , , 
C1
3-4, 75, 105
divides, divisibility, divisor, multiple
greatest common divisor
prime, composite
coprime, relatively prime
Division Algorithm
quotient, remainder
congruence
C1, G5-6
C2, G7
C2
C2
C2, G5
C2
C3
17, 47
49
22, 27
27
15, 45, 197
15, 197
31
n, modular arithmetic
C3
34-37
F[x], R[x], polynomial ring
Mn(R), ring of n-by-n matrices over R
C1, G4
C1
C2, E1
D2, E1
D2, E1
D3
D3, E1-2
42
79
144
[√𝑑 ]
conjugate
norm
unique factorization
units of a ring R, R× = U(R)
Textbook pages
146
24
103, 317
associate
reducible, irreducible, factoring
migration of units
operation (unary, binary, n-ary)
ring
zero element (additive identity)
additive inverse
commutative ring
ring with identity (unity)
isomorphism, isomorphic
field
zero divisor
integral domain (domain)
Euclidean domain
subring
quotient field
D4
D4, E3,G8
D4
F1
F1
F1
F1
F3
F3
F4
F4
F4
F5, F6
F5
F5
F6, G10
, real quaternions
F6-7
, octonions, nonassociative
F7
skewfield (division ring)
R[[t]], ring of power series
coefficient
degree of a polynomial
zero (root) of a polynomial
monic polynomial
F(t), field of rational functions (rational expressions)
F((t)), Laurent series
proper subring
homomorphism
image, kernel
automorphism
ideal, left/right ideal
principal ideal, generator
principal ideal ring
coset
quotient ring
well-defined map
canonical homomorphism (natural homomorphism)
epimorphism, monomorphism
transcendental, algebraic
First Isomorphism Theorem for Rings
F7
G1
G4
G4
G5
G7
G10
G10
G13
H1
H1
H3
H9
H10
H11
H12
H12-13
H13
H14
H3
H12
H15
58, 141
55, 56
73
74
74
74
78
93
247
105
101
102
197
89
42
43
48
195
90
211
226
158
160
237
240
35, 237
241
493, 527
250