Download 1. This is significantly longer than an actual final problem, in order to

1. This is significantly longer than an actual final problem, in order to give
you more possible definitions:
(a) Define the following: Groups, rings, modules, fields, homomorphisms,
ideals, subrings, quotient groups, quotient rings, (commutative) integral domains, UFDs, PIDs, Euclidean domains, algebraic numbers,
algebraic integers, wallpaper groups, frieze groups, Cn , Dn , Sn , An ,
R[x1 , . . . , xn ] where R is a ring, Fp , Fpk , maximal ideals, prime
elements, irreducible elements, associates, units, linear transformations, point groups, Jordan Normal Form matrices, row-echelon and
reduced row-echelon form matrices, permutation matrices, determinant, sign of a permutation, left and right cosets, equivalence relations.
• Groups: A group G is a monoid for which, for every element a ∈ G,
there exists b ∈ G such that ab = e = ba.
• Rings: A ring (R, +, ·), is a set R equipped with two operations,
called addition + and multiplication ·, such that (R, +) is an abelian
group and (R, ·) is a monoid.
• Modules: A module over a ring R is an abelian group M equipped
with an action RM → M, (r, m) 7→ r · m, satisfying the following
axioms ∀r, s ∈ R, m, n ∈ M .
(rs)(m) = r(sm)
1(m) = m
(r + s)(m) = rm + sm
r(m + n) = rm + rn
• Fields: A field is a commutative division ring.
• Homomorphisms: A homomorphism is a map f : X → Y such that
f (ab) = f (a)f (b) ∀ a, b ∈ X.
• Ideals: An ideal I of a ring R is a nonempty subset of R with these
properties
– I is closed under addition, and
– If s ∈ I and r ∈ R then rs ∈ I
• Subrings: A subring of a ring R is a subset that is closed under
the operations of addition, subtraction, and multiplication and that
contains the element 1.
• Quotient Groups: The quotient group G/N is the set of cosets of N
in G.
• Quotient Rings: In general, a quotient ring is a set of equivalence
classes where [x] = [y] ⇔ x − y ∈ a.
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• (commutative) Integral Domains: An integral domain R is a ring with
this property: R is not the zero ring, and if a and b are elements of
R whose product ab is zero, then a = 0 or b = 0.
• UFDs: An integral domain R is a unique factorization domain if it
has these properties:
– Factoring terminates
– The irreducible factorization of an element a is unique
• PIDs: An integral domain in which every ideal is principal is called
a principal ideal domain
• Euclidean Domains: A euclidean domain is a principal ideal domain.
• Algebraic Numbers: A complex number α that is the root of a polynomial with rational coefficients is called an algebraic number
• Algebraic Integers: An algebraic number is an algebraic integer if its
(monic) irreducible polynomial over Q has integer coefficients.
• Wallpaper Groups: A wallpaper group is the classification of a twodimensional repetitive pattern based on the symmetries in the pattern
• Frieze Groups: A frieze group is the classification of designs on twodimensional surfaces which are repetitive in one direction, based on
the symmetries in the pattern
• Cn : The cyclic group of rotations by multiples of 2π/n about a line,
with n arbitrary
• Dn : The dihedral group of symmetries of a regular n-gon, with n
arbitrary
• Sn : The symmetric group Sn is the group of permutations of the
indices {1, 2, . . . , n}
• An : The alternating group An is the group of even permutations
• R[x1 , . . . , xn ] where R is a ring is the ring of polynomials with coefficients in R
• Fp is the finite field of the integers modulo some prime p.
• Fpk is a finite field of order pk where p is some prime and k is a
positive integer, and the characteristic is p.
• Maximal ideals: A maximal ideal M of a ring R is an ideal that isn’t
equal to R, and that isn’t contained in any ideal other than M and
R: If an ideal I contains M , then I = M or I = R.
• Prime elements: p is a prime element if p is not a unit, and whenever
p divides a product ab, then p divides a or p divides b.
• Irreducible elements: a is irreducible is a is not a unit, and it has no
proper divisor - its only divisors are units and associates.
• Associates: a and b are associates ⇔ (a) = (b)
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• Units: u is a unit ⇔ (u) = (1)
• Linear transformations: A linear transformation T V → W from one
vector space over a field F to another is a map that is compatible
with addition and scalar multiplication:
T (v1 + v2 ) = T (v1) + T (v2 )andT (cv1 ) = cT (v1)
for all v1 and v2 in V and all c in F .
• Point groups: The point group G is the image of G in the orthogonal
group O2 .
• Jordan Normal Form Matrix: A Jordan Normal Form matrix of an
n × n matrix A is another n × n matrix made of m × m blocks of the
form
.
λ = λ 1λ 1 . . 1λ 1λ ,
1
1
1
1
1
where λ1 produces m − 1 generalized eigenvectors (and 1 regular
eigenvector), and is of the form
λ1 λ2
..
. λj
where λ1 , . . . , λj are the eigenvalues of the A.
• Row-echelon Matrix: A matrix is in row-echelon form if:
– All nonzero rows (rows with at least one nonzero element) are
above any rows of all zeroes (all zero rows, if any, belong at the
bottom of the matrix).
– The leading coefficient (the first nonzero number from the left,
also called the pivot) of a nonzero row is always strictly to the
right of the leading coefficient of the row above it
• Reduced row-echelon Matrix: A matric is in reduced row-echelon
form if:
– It is in row echelon form.
– Every leading coefficient is 1 and is the only nonzero entry in its
column.
• Permutation Matrix: There is a permutation matrix P associated to
any permutation p. Left multiplication by this permutation matrix
permutes the entries of a vector X using the permutation p.
– A permutation matrix P always has a single 1 in each row and
in each column, the rest of its entries being 0.
– The determinant of a permutation matrix is ±1
• Determinant: The determinant of an n × n matrix is a function det:
Rn×n → R
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• Sign of a permutation: The sign of a permutation σ can be defined
from its decomposition into the product of transpositions as
sgn(σ) = (−1)m
where m is the number of transpositions in the decomposition.
• Left coset: If H is a subgroup of G and if a is an element of G, the
subset
aH = {ah : h ∈ H}
is called a left coset.
• Right coset: The right cosets of a group G are the sets
Ha = {ha : h ∈ H}
• Equivalence Relations: An equivalence relation on a set S is a relation
that holds between certain pairs of elements of S, written as a ∼ b.
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