Rings and Modules

Question 1

Define a ring

Answer 1

A ring is a datum (R, +, ×, 0, 1) where R is a set, 1, 0 ∈ R and + and x are binary operations on R such that:
1. R is an abelian group under + with identity element 0.
2. R is a monoid under x, that is x is associative and has identity 1.
3. Multiplication distributes over addition: x(y + z) = xy + xz, (x + y)z = xz + yz, for all x, y, z ∈ R.

Question 2

Define a commutative ring

Answer 2

A ring where x is commutative.

Question 3

Define the direct sum of rings

Answer 3

Let R and S be rings, then the ring R⊕S is given by the pairs (r, s), r ∈ R and s ∈ S and addition and multiplication are given component-wise.

Question 4

Define a subring

Answer 4

If R is a ring, a subset S ⊆ R is a subring if 0, 1 ∈ S and S is closed under the addition and multiplication operations in R.

Question 5

Define a ring homomorphism

Answer 5

A map f: R → S between rings R and S is a ring homomorphism if:
1. f(1_R) = 1_S
2. f(r1 + r2) = f(r1) + f(r2)
3. f(r1.r2) = f(r1). f(r2)

Question 6

Define the image of a ring homomorphism

Answer 6

Given a ring homomorphism f: R→S, im(f) = {s ∈ S: ∃r ∈ R such that f(r) = s}.

Question 7

Define the kernel of a ring homomorphism

Answer 7

Given a ring homomorphism f: R→S, ker(f) = {r ∈ R: f(r) = 0}.

Question 8

Define a ring isomorphism

Answer 8

A homomorphism f: R→S is an isomorphism if there exists another homomorphism g: S→R such that f◦g = id_S and g◦f = id_R.

Question 9

Define the characteristic of a ring

Answer 9

The minimum d such that 1+1+…+1 (d times) = 0 if such a d exists, otherwise the characteristic is 0.

Question 10

Define a zero divisor

Answer 10

If R is a ring, then an element a ∈ R{0} is said to be a zero-divisor if there is some b ∈ R{0} such that a.b = 0.

Question 11

Define an integral domain

Answer 11

An integral domain is a ring which is not the zero ring and has no zero-divisors is called an integral domain.

Question 12

Define a unit

Answer 12

Let R be a ring. The subset R× = {r ∈ R : ∃s ∈ R such that r.s = 1}, is called the group of units in R.

Question 13

Define an ideal

Answer 13

Let R be a ring. A subset I ⊆ R is called an ideal if it is a subgroup of (R, +) and for any a ∈ I and r ∈ R, a.r ∈ I.

Question 14

Define a generated ideal

Answer 14

Given any subset T of R, the ideal generated by T is given by <T> = intersection of all I (where I is an ideal) such that T⊆I.</T>

Question 15

Define a principal ideal

Answer 15

An ideal generated by a single element.

Question 16

Define associate elements

Answer 16

Two elements a, b ∈ R are said to be associates if there is a unit u ∈ R× such that a = u.b.

Question 17

Give the universal property of quotients

Answer 17

Suppose that R is a ring, I is an ideal of R, and q: R → R/I the quotient homomorphism. If ϕ: R → S is a ring homomorphism such that I ⊆ ker(ϕ), then there is a unique ring homomorphism ϕ’: R/I → S such that ϕ’◦q = ϕ. Moreover ker(ϕ’) is the ideal ker(ϕ)/I = {m + I : m ∈ ker(ϕ)}.

Question 18

Give the first isomorphism theorem of rings

Answer 18

If ϕ: R → S is a homomorphism then ϕ induces an isomorphism ϕ’: R/ker(ϕ) → im(ϕ).

Question 19

Give the second isomorphism theorem of rings

Answer 19

If R is a ring, A is a subring of R and I an ideal of R, then (A + I)/I is isomorphic to A/(A ∩ I)

Question 20

Give the third isomorphism theorem of rings

Answer 20

Suppose that I1 ⊆ I2 are ideals in R. Then (R/I1)/(I2/I1) is isomorphic to R/I2.

Question 21

Give the Chinese remainder theorem for rings

Answer 21

Let R be a ring, and I, J ideals of R such that I + J = R. Then R/(I∩J) is isomorphic to R/I⊕R/J.

Question 22

Define a maximal ideal

Answer 22

Let R be a ring, and I an ideal of R. I is a maximal ideal of R if it is not strictly contained in any proper ideal of R.

Question 23

Define a prime ideal

Answer 23

Let R be a ring, and I an ideal of R. I is a prime ideal of R if I ≠ R and for all a, b ∈ R, whenever a.b ∈ I then either a ∈ I or b ∈ I.

Question 24

Define a prime element

Answer 24

If a prime ideal I is principal any generator of I is said to be a prime element of R.

Question 25

Define the degree of a polynomial

Answer 25

If R is a ring and f ∈ R[t] is nonzero, then we may write f = the sum over i from 0 to n of (a_i)(t^i), where a_n ≠ 0. Then, the degree deg(f) of f is n.

Question 26

Give the division algorithm

Answer 26

Let R be a ring and f = the sum over i from 0 to n of (a_i)(t^i) ∈ R[t], where a_n ∈ R×. Then if g ∈ R[t] is any polynomial, there are unique polynomials q, r ∈ R[t] such that either r = 0 or deg(r) < deg(f) and g = q.f + r.

Question 27

Define a Euclidean Domain

Answer 27

Let R be an integral domain. R is a Euclidean domain if there exists a function N: R\{0} → Natural numbers such that given any a, b ∈ R with b ≠ ,0 there are q,r ∈ R such that a = b.q + r and either r = 0 or N(r) < N(b).

Question 28

Define a principal ideal domain

Answer 28

A principal ideal domain is an integral domain in which every ideal is principal.

Question 29

Define an irreducible element

Answer 29

Let R be an integral domain. A nonzero element r ∈ R is said to be irreducible if whenever r = a.b then exactly one of a or b is a unit.

Question 30

Define a field extension

Answer 30

If E, F are any fields and F ⊆ E we call E a field extension of F and write E/F.

Question 31

Define the degree of a field extension

Answer 31

If E is finite dimensional as an F-vector space, we write [E : F] = dim(E) for this dimension and call it the degree of the field extension E/F.

Question 32

State the tower law

Answer 32

Let F ⊂ E ⊂ K be fields, then [K : F] is finite if and only if both degrees [E : F], [K : E] are, and when they are finite we have [K : F] = [E : F][K : E]

Question 33

Define an algebraic number

Answer 33

Let α ∈ C. We say that α is algebraic over Q if there is a field E which is a finite extension of Q containing α.

Question 34

Define a transcendental number

Answer 34

Let α ∈ C. We say that α is transcendental over Q if there are no fields E which are a finite extension of Q containing α.

Question 35

Define the minimal polynomial

Answer 35

Given α ∈ C, the irreducible polynomial f for which Q(α) is isomorphic to Q[t]/⟨f⟩ is called the minimal polynomial of α over Q.

Question 36

Define a factor

Answer 36

Let R be an integral domain. If a, b ∈ R we say that a is a factor of b, and write a|b, if there is some c ∈ R such that b = a.c.

Question 37

Define a common factor

Answer 37

Let R be an integral domain. We say c ∈ R is a common factor of a, b ∈ R if c|a and c|b.

Question 38

Define the highest common factor

Answer 38

c is the highest common factor of a and b if whenever d is a common factor of a and b, d|c.

Question 39

Define a common multiple

Answer 39

Let R be an integral domain. We say that k ∈ R is a common multiple if a|k and b|k.

Question 40

Define the lowest common multiple

Answer 40

The least common multiple is a common multiple which is a factor of every common multiple.

Question 41

Define a unique factorisation domain

Answer 41

An integral domain R is said to be an unique factorisation domain if every element of R\{0} is either a unit, or can be written uniquely up to reordering and units as a product of irreducible elements.

Question 42

Define a Noetherian ring

Answer 42

A ring is Noetherian if any nested ascending chain of ideals stabilises.

Question 43

Define content in the integer field

Answer 43

If f ∈ Z[t] then the content c(f) of f is the highest common factor of the coefficients of f.

Question 44

State Gauss' content lemma

Answer 44

Let f, g ∈ Z[t]. Then c(f.g) = c(f).c(g).

Question 45

Define content in the rational field

Answer 45

Suppose f ∈ Q[t] is nonzero. Then there is a unique α ∈ Q>0 such that f = αf′ where f′ ∈ Z[t] and c(f′) = 1. We write c(f) = α.

Question 46

State Eisenstein's criterion

Answer 46

Suppose that f ∈ Z[t] has c(f) = 1, and f = (a_n)(t^n) + a_(n−1)t^(n−1) + … + (a_1)t + a_0. Then if there is a prime p ∈ Z such that p|ai for all i, 0 ≤ i ≤ n − 1 but p does not divide an and p^2 does not divide a0 then f is irreducible in Z[t] and Q[t].

Question 47

Define a module

Answer 47

Let R be a ring with identity 1. A module over R is an abelian group (M, +) and R’s multiplication action a: R x M → M, written (r, m) → r.m satisfying: 1. 1.m = m, for all m ∈ M 2. (r1.r2).m = r1.(r2.m), for all r1,r2 ∈ R, m ∈ M 3. (r1 + r2).m = r1.m + r2.m for all r1,r2 ∈ R and m ∈ M 4. r.(m1 + m2) = r.m1 + r.m2 for all r ∈ R and m1, m2 ∈ M

Question 48

Define a submodule

Answer 48

If M is an R-module, a subset N ⊆ M is called a submodule if it is an abelian subgroup of M and whenever r ∈ R and n ∈ N then r.n ∈ N.

Question 49

Define a generated module

Answer 49

If X is any subset of an R-module M then the submodule generated by X is defined to be: ⟨X⟩ = Intersection of all submodules N of M containing X.

Question 50

Define a linearly independent set

Answer 50

If M is a module over R, we say a set S ⊆ M is linearly independent if whenever we have an equation r1.s1 + r2.s2 + … + rk.sk = 0 for ri ∈ R, si ∈ S, then r1 = r2 = . . .rk = 0.

Question 51

Define a basis

Answer 51

We say that a set S is a basis for a module M if and only if it is linearly independent and it generates M.

Question 52

Define a free module

Answer 52

Any module which has a basis is called a free module.

Question 53

Define a finitely generated module

Answer 53

A module is finitely generated if it is generated by a finite subset.

Question 54

Define a quotient module

Answer 54

Let N is a submodule of M, and if r ∈ R and m + N ∈ M/N then define r.(m + N) = r.m + N. This module M/N is called the quotient module of M by N.

Question 55

Define a module homomorphism

Answer 55

If M1, M2 are R-modules, then ϕ: M1 → M2 is an R-module homomorphism if: ϕ(m1 + m2) = ϕ(m1) + ϕ(m2) for all m1, m2 ∈ M1 and ϕ(r.m) = r.ϕ(m), for all r ∈ R, m ∈ M1.

Question 56

Define an element's annhilator

Answer 56

Let M be an R-module and suppose that m ∈ M. Then the annihilator of m is {r ∈ R : r.m = 0}.

Question 57

Define a torsion element

Answer 57

Let M be an R-module and suppose that m ∈ M. Then, m is a torsion element if AnnR(m) is nonzero.

Question 58

Define a torsion module

Answer 58

A module M is torsion if every m ∈ M is a torsion element.

Question 59

Define a torsion-free module

Answer 59

A module M is torsion if there are no non-zero torsion elements.

Question 60

Define a cyclic module

Answer 60

A cyclic module is a module generated by a single element.

Question 61

Define the rank of a module

Answer 61

Let M be a finitely generated free R-module. Then the size of a basis of M is the rank rk(M) of M.

Question 62

Define an elementary row operation

Answer 62

Let A ∈ Mm,n(R) be a matrix, and let r1,r2, … ,rm be the rows of A, which are row vectors in R^n. An elementary row operation on a matrix A ∈ Mm,k(R) is an operation of the form: 1. Swap two rows ri and rj . 2. Replace one row, row i say, with a new row r′i = ri + crj for some c ∈ R, and j, i.

Question 63

Define equivalent matrices

Answer 63

If A, B ∈ Mn,m(R) we say that A and B are equivalent if B = PAQ where P ∈ Mn,n(R) and Q ∈ Mm,m(R) are invertible matrices.

Question 64

Give the Smith Normal Form Theorem

Answer 64

Suppose that A ∈ Mn,m(R) is a matrix. Then A is equivalent to a diagonal matrix D where if k = min{m, n} then there are d1, d2, …, dk on the diagonal, where d1|d2|…|dk and this choice is unique up to units.

Question 65

Define a presentation

Answer 65

Let M be a finitely generated R-module. The pair of maps ϕ, ψ defined such that im(ϕ) = M and ψ: R^m → R^n has image im(ψ) = ker(ϕ) is called a presentation of the finitely generated modules M.

Question 66

Define a resolution

Answer 66

A presentation where the map ψ, where im(ψ) = ker(ϕ) for ϕ such that im(ϕ) = M can be chosen to be injective is called a resolution of the module M.

Question 67

State the structure theorem

Answer 67

Suppose that M is a finitely generated module over a Euclidean domain R. Then there is an integer s and nonzero non-units c1, c2,… , cr ∈ R, where c1|c2| . . . |cr and M is isomorphic to (The direct sum from 1 to r of R/ciR) ⊕ R^s.

Question 68

State the structure theorem of finitely generated abelian groups

Answer 68

Let A be a finitely generated abelian group. Then there exists an integer r ∈ Z≥0 and integers c1, c2,… , ck ∈ Z greater than 1 such that c1|c2| . . . |ck and A is isomorphis to Z^r ⊕ (Z/c1Z) ⊕ … ⊕ (Z/ckZ). Moreover the integers r, c1, . . . , ck are uniquely determined.

Question 69

State the structure theorem in primary decomposition form

Answer 69

Let R be a Euclidean domain and suppose that M is a finitely generated R-module. Then there are irreducibles p1,… , pk ∈ R and integers s, ri for 1 ≤ i ≤ k, such that: M is isomorphic to the direct sum from 1 to k of (R/pi^(ri)R) ⊕ R s . Moreover, the pairs (pi ,ri) are uniquely determined up to units (where the units act on the pi only).

Question 70

Define the companion matrix

Answer 70

For a monic polynomial f = t^n + a_(n-1)t^(n−1) + … + a_0 of degree n ≥ 1, the n × n matrix is given by the identity shifted down one row and the final column being -a_(i-1) for row i.

Question 71

State the rational canonical form theorem

Answer 71

Suppose that V is a nonzero finite dimensional k-vector space and T: V → V is a linear map. Then there are unique nonconstant monic polynomials f1,…, fk ∈ k[t] such that f1| f2| . . . | fk and a basis of V with respect to which T has matrix which is block diagonal with blocks C(fi).