Definitions

Question 1

Commutative Ring

Answer 1

A commutative ring with 1 is an algebraic structure (R, +, •, 0, 1) such that 1=!0, 0,1 € R so |R| >= 2
(R, +, 0) is an abelian group
• is associative and commutative
R is distributive

Question 2

Invertible

Answer 2

Let R be a ring and let x € R. We say x is invertible if xy=1 for some y € R

Question 3

Domain

Answer 3

Let R be a ring and x,y€R, we say R is a domain if xy=0 => x=0 or y=0

Question 4

Field

Answer 4

R is a field if R^x = R{0}

Any field is a domain

Question 5

R^x

Answer 5

R^x = {x € R| x is invertible in R}

This is called the group of units

Question 6

Ring of Polynomials

Answer 6

Let R be a ring and X be a variable. The polynomial ring R[X] consists of all formal expressions
a_n X^n + … + a_1 X + a_0 with a_i € R

Question 7

Monomial

Answer 7

The polynomial X^i = 1•X^i

(i € Z_>=0, X^0=1) are called monomials

Question 8

Unit

Answer 8

x is a unit of R if x has an inverse in R

Question 9

Inverse

Answer 9

y is an inverse of x if xy = 1

Question 10

Ideal

Answer 10

A subset I ⊆ R is an ideal of R if the following
properties hold:
(i) 0 ∈ I
(ii) a, b ∈ I ⇒ a + b ∈ I
(iii) a ∈ I, r ∈ R ⇒ ra ∈ I

Question 11

The ideal generated by A

Answer 11

Let A ⊆ R. The ideal generated by A is
.A. = SUM_a∈A (r_a a) : r_a ∈ R, all but finitely many zero)
= {r1a1 + · · · + rnan : n ∈ N, a1, . . . , an ∈ A, r1, . . . , rn ∈ R} .
We also call A a generating set for .A.

Question 12

Principal Ideal

Answer 12

An ideal I of R is principle if I = .a. for some a ∈ R.

Question 13

Principle Ideal Domain

Answer 13

A PID is a domain in which every ideal is principle

Question 14

Euclidean Norm

Answer 14

A euclidean norm on R is a map N : R \ {0} → N such that for any
a, b ∈ R \ {0} there exist q, r ∈ R with a = qb + r and either r = 0 or N(r) < N(b). The
ring R is called euclidean if there is some euclidean norm on R.

Question 15

Maximal ideal

Answer 15

. Let I be an ideal of a
ring R. The ideal I is maximal if it is proper and there is no ideal J with I $ J $ R.
The ideal I is prime if for a, b ∈ R, ab ∈ I implies a ∈ I or b ∈ I.

Question 16

Noetherian

Answer 16

A ring R is noetherian if every ideal of R is finitely generated.

Question 17

a divides b

Answer 17

Let a, b ∈ R.

a divides b, a|b if b ∈ Ra

Question 18

associated

Answer 18

a and b are associated (written a ∼ b) if there is a unit u ∈ R× such that b = ua

Question 19

Irreducible

Answer 19

p is irreducible if p /∈ R× and if p = xy with x, y ∈ R, then x ∈ R× or y ∈ R×

Question 20

Prime

Answer 20

p is prime if p !∈ R× and if p|xy for x, y ∈ R, then p|x or p|y

Question 21

Unique Factorisation Domain

Answer 21

A unique factorization domain (or

UFD) is a domain R in which every 0 != x ∈ R \ R× is a product of prime elements

Question 22

Primitive

Answer 22

A polynomial f ∈ R[X] is primitive if the coefficients of f have no common divisor that is not a unit

Question 23

Root

Answer 23

An element α ∈ F is a root of f if f(α) = 0.

Question 24

Integrally Closed

Answer 24

A domain R is integrally closed if the following holds: If f ∈ R[X] is monic and α ∈ Frac(R) is a root of f, then α ∈ R.

Question 25

Eisenstein

Answer 25

Suppose that F = Frac(R). The polynomial f = anX^n + · · · + a1X + a0 ∈ R[X] is Eisenstein with respect to the prime p ∈ R if p!|an, p|an−1, . . . , p|a0, p2!|a0.

Question 26

Partial Order

Answer 26

A partial order on a set S is a binary relation <= which is (i) reflexive: For x ∈ S, x <= x. (ii) antisymmetric: For x, y ∈ S, if x <= y and y <= x, then x = y. (iii) transitive: For x, y, z ∈ S, if x <= y and y <= z, then x <= z.

Question 27

Monomial Ordering

Answer 27

A monomial ordering is a partial order <= on M = M(X1, . . . , Xn) that satisfies: (i) <= is total (also called linear), i.e. if m, m0 ∈ M then m <= m0 or m0 <= m. (ii) 1 <= m for every m ∈ M. (iii) For every m1, m2, m ∈ M, if m1 <= m2, then mm1 <= mm2.

Question 28

Lexicographic Order

Answer 28

For α, β ∈ N^n we define X^α ≺Lex X^β iff the | leftmost non-zero component of α − β is negative.

Question 29

Degree Lexicographic Order

Answer 29

For α, β ∈ N^n we define X^α ≺Deglex X^β iff either deg(X^α) < deg(X^β) or deg(X^α) = deg(X^β) and X^α ≺Lex X^β .

Question 30

Well Ordering

Answer 30

A total order <= on a set S is a well-ordering if every | non-empty subset of S has a least element with respect to <=.

Question 31

Polynomial Ring

Answer 31

A polynomial in variables X1, . . . , Xn with coefficients in R is an expression of the form f = a1m1 + · · · + akmk, where a1, . . . , ak ∈ R, m1, . . . , mk ∈ M(X1, . . . , Xn)

Question 32

lm_<=(f)

Answer 32

lm_<=(f) = m1, the leading monomial of f

Question 33

lc_<=(f)

Answer 33

lc_<=(f) = a1, the leading coefficient of f

Question 34

lt_<=(f)

Answer 34

lt_<=(f) = a1m1, the leading term of f

Question 35

monic

Answer 35

f is monic if lc_<=(f) = 1

Question 36

deg(f)

Answer 36

deg(f) = max{deg(m1), . . . deg(mk)}, the total degree of f

Question 37

f reduces to h modulo g in one step

Answer 37

Let f, g, h ∈ K[X] and write f = a1m1 +· · ·+akmk in standard form. We say that f reduces to h modulo g in one step (with respect to <=), written f→ h, if lm_<=(g) divides a monomial mi of f

Question 38

Monomial Ideal

Answer 38

The ideal I of K[X] is a monomial ideal if it is generated by a set of monomials.

Question 39

Grobner Basis

Answer 39

A set G ⊆ I \ {0} is a Grobner basis of I with respect | to <= if G is finite and LT_<=(I) = ..

Question 40

Reduced Grobner Basis

Answer 40

A Grobner basis G = {g1,..,gk} of I is reduced if each gi is monic and reduced modulo every other gj, j != i.

Question 41

Radical

Answer 41

Let I be an ideal of a ring R. The radical of I is √I = {a ∈ R : a^n ∈ I for some n ∈ N}. The ideal I is a radical ideal if I = √I.