Basics

Finite rings and their elements implement the ring interface. We document the methods that are unique to finite rings.

Creation of finite rings

finite_ring Method

finite_ring(c::Vector{IntegerUnion},
            m::Vector{ZZMatrix}; check::Bool = true)

Return the finite ring with additive group isomorphic to $\mathbf{Z}/c_1\mathbf{Z}\times \cdots \times \mathbf{Z}/c_n\mathbf{Z}$ and such that generators satisfy $b_i{b}_j=\sum _{k=1}^n{m}_{jk}^{(i)}{b}_k$, where $m^{(i)}$ is the $j$-th entry of $m$.

If check is true (default), it is verified that this defines a ring.

Examples

julia> finite_ring([2, 2], [ZZ[1 0; 0 0], ZZ[0 0; 0 1]]) # Z/2Z x Z/2Z
Finite ring with additive group
  isomorphic to (Z/2)^2
  and with 2 generators and 2 relations

source

finite_ring Method

finite_ring(c::Vector{IntegerUnion},
            m::Vector{ZZMatrix}; check::Bool = true)

Return the finite ring with additive group isomorphic to $\mathbf{Z}/c_1\mathbf{Z}\times \cdots \times \mathbf{Z}/c_n\mathbf{Z}$ and such that generators satisfy $b_i{b}_j=\sum _{k=1}^n{m}_{jk}^{(i)}{b}_k$, where $m^{(i)}$ is the $j$-th entry of $m$.

If check is true (default), it is verified that this defines a ring.

Examples

julia> finite_ring([2, 2], [ZZ[1 0; 0 0], ZZ[0 0; 0 1]]) # Z/2Z x Z/2Z
Finite ring with additive group
  isomorphic to (Z/2)^2
  and with 2 generators and 2 relations

source

finite_ring Method

finite_ring(A::AbstractAssociativeAlgebra) -> FiniteRing, FiniteRingMap

Given an algebra $A$ over a finite field $k$, return $(R,f)$, where $R$ is a finite ring and $f:R\to A$ an isomorphism rings. Currently, the field $k$ must be a prime field.

Examples

julia> R, f = finite_ring(matrix_algebra(GF(2), 3));

julia> R
Finite ring with additive group
  isomorphic to (Z/2)^9
  and with 9 generators and 9 relations

source

finite_ring Method

finite_ring(A::MatRing{FiniteRingElem}) -> FiniteRing

Given a matrix ring $A$ over a finite ring, return $(R,f)$, where $R$ is a finite ring and $f:R\to A$ an isomorphism rings. Currently, the field $k$ must be a prime field.

Examples

julia> R, = finite_ring(matrix_algebra(GF(2), 2));

julia> S = matrix_ring(R, 2);

julia> T, = finite_ring(S);

julia> T
Finite ring with additive group
  isomorphic to (Z/2)^16
  and with 16 generators and 16 relations

source

Properties

additive_generators Method

additive_generators(R::FiniteRing) -> Vector{FiniteRingElement}

Return generators of the additive group of a finite ring $R$.

Examples

julia> R, _ = finite_ring(matrix_algebra(GF(2), 2));

julia> additive_generators(R)
4-element Vector{FiniteRingElem}:
 Finite ring element [1, 0, 0, 0]
 Finite ring element [0, 1, 0, 0]
 Finite ring element [0, 0, 1, 0]
 Finite ring element [0, 0, 0, 1]

source

number_of_additive_generators Method

number_of_additive_generators(R::FiniteRing) -> Int

Return the number generators of the additive group of a finite ring $R$.

Examples

julia> R, _ = finite_ring(matrix_algebra(GF(2), 2));

julia> number_of_additive_generators(R)
4

source

elementary_divisors Method

elementary_divisors(R::FiniteRing) -> Vector

Return the elementary divisors of the additive group of the finite ring $R$.

Examples

julia> R, f = finite_ring(matrix_algebra(GF(2), 2));

julia> elementary_divisors(R)
4-element Vector{ZZRingElem}:
 2
 2
 2
 2

source

Maximal quotient rings

maximal_p_quotient_ring Method

maximal_p_quotient_ring(R::FiniteRing, p::IntegerUnion) -> FiniteRing, FiniteRingHom

Given a finite ring $R$ and a prime $p$, return the largest quotient ring $S$ of $p$-power order and the projection $R\to S$.

Examples

julia> R = finite_ring([6, 6], [ZZ[1 0; 0 0], ZZ[0 0; 0 1]]); # Z/6Z x Z/6Z

julia> S, f = maximal_p_quotient_ring(R, 3);

julia> S
Finite ring with additive group
  isomorphic to (Z/3)^2
  and with 2 generators and 4 relations

source

Decomposition

central_primitive_idempotents Method

central_primitive_idempotents(R::FiniteRing) -> Vector{FiniteRingElem}

Given a finite ring $R$, return the set of orthgonal central primitive idempotents of $R$.

Examples

julia> R = finite_ring([6, 6], [ZZ[1 0; 0 0], ZZ[0 0; 0 1]]); # Z/6Z x Z/6Z

julia> central_primitive_idempotents(R)
4-element Vector{FiniteRingElem}:
 Finite ring element [0, 3]
 Finite ring element [3, 0]
 Finite ring element [4, 0]
 Finite ring element [0, 4]

source

decompose_into_indecomposable_rings Method

decompose_into_indecomposable_rings(R::FiniteRing)
                                -> Vector{FiniteRing}, Vector{FiniteRingHom}

Given a finite ring $R$, return a list of indecomposable rings $S$ and a list of projections $R\to S$, such that $R$ is the direct product of these rings.

Examples

julia> R = finite_ring([6, 6], [ZZ[1 0; 0 0], ZZ[0 0; 0 1]]); # Z/6Z x Z/6Z

julia> rings, projections = decompose_into_indecomposable_rings(R);

julia> length(rings)
4

source

is_indecomposable Method

is_indecomposable(R::FiniteRing) -> Bool

Return whether the finite ring $R$ is indecomposable.

Examples

julia> R = finite_ring([6, 6], [ZZ[1 0; 0 0], ZZ[0 0; 0 1]]); # Z/6Z x Z/6Z

julia> is_indecomposable(R)
false

julia> R, _ = finite_ring(matrix_algebra(GF(2), 2));

julia> is_indecomposable(R)
true

source

Creaton of elements

Elements of finite rings can be constructed by specifying the coordinates with respect to the additive additive generators or as linear combinations of the additive generators:

julia> R = finite_ring([6, 6], [ZZ[1 0; 0 0], ZZ[0 0; 0 1]]); # Z/6Z x Z/6Z

julia> x = R([1, 2])
Finite ring element [1, 2]

julia> e1, e2 = additive_generators(R);

julia> y = e1 + 2 * e2;

julia> x == y
true

Conversion to other structures

Finite rings of prime characteristic can be converted to algebras using the isomorphism method:

julia> R, = finite_ring(GF(2)[small_group(8, 3)]);

julia> S = matrix_ring(R, 2);

julia> T, = finite_ring(S);

julia> f = isomorphism(MatAlgebra, R)
Map
  from finite ring
  to matrix algebra of dimension 8 over Prime field of characteristic 2

julia> f = isomorphism(StructureConstantAlgebra, R)
Map
  from finite ring
  to structure constant algebra of dimension 8 over GF(2)