Elementary Operations on Subalgebras and Ideals

Bases

The functions described here assume that the matrix algebra \(R\) is defined over a ring \(S\) with a matrix echelonization algorithm. Magma computes a basis for \(R\) considered as a \(S\)-module when necessary so then operations like membership testing can be performed. The following functions allow one to access this basis.

Dimension(R): AlgMatV -> RngIntElt

Assuming that \(R\) is a subalgebra of \(M_n(S)\), return the dimension of \(R\), considered as a \(S\)-module.

Basis(R): AlgMatV -> [ AlgMatElt ]

Assuming that \(R\) is a subalgebra of \(M_n(S)\), return the \(S\)-basis of \(R\), considered as a \(S\)-module. The basis is returned as a sequence of matrices of \(R\).

BasisElement(R, i): AlgMatV, RngIntElt -> AlgMatElt

Given \(R\) a subalgebra of \(M_n(S)\), return the \(i\)-th element of the \(S\)-basis of \(R\), where \(i\) must be between 1 and the dimension of \(R\).

Coordinates(R, X): AlgMatV, AlgMatVElt -> [ RngElt ]

Assuming that \(R\) is a subalgebra of \(M_n(S)\), and given an element \(X\) of \(R\), return the coordinates of \(X\) with respect to the basis of \(R\). If \(R\) has dimension \(k\) over its coefficient ring \(S\), and \(R\) has basis \(U_1, \ldots, U_k\), the coordinates are returned as the unique sequence \([a_1, \ldots, a_k]\) of elements of \(S\) such that \(X = a_1 U_1 + \ldots + a_r U_r\).

Intersection of Subalgebras

R meet T: AlgMat, AlgMat -> AlgMat

Given algebras \(R\) and \(S\) that are subalgebras of the same complete algebra \(M_n(S)\), where \(S\) is a PIR, this operator constructs their intersection.

Membership and Equality

The operations described here assume that the matrix algebra is defined over a principal ideal ring.

x in R: AlgMatElt, AlgMat -> BoolElt

X subset R: { AlgMatElt}, AlgMat -> BoolElt

T subset R: AlgMat, AlgMat -> BoolElt

Given a matrix \(x\) (set of matrices \(X\), matrix algebra \(T\)) and a matrix algebra \(R\) all belonging to a common matrix algebra defined over a PIR, return true if \(x\) (\(X\), \(T\), respectively) is contained in \(R\), false otherwise.

x notin R: AlgMatElt, AlgMat -> BoolElt

X notsubset R: { AlgMatElt}, AlgMat -> BoolElt

T notsubset R: AlgMat, AlgMat -> BoolElt

Given a matrix \(x\) (set of matrices \(X\), matrix algebra \(T\)) and a matrix algebra \(R\) all belonging to a common matrix algebra defined over a PIR, return true if \(x\) (\(X\), \(T\), respectively) is not contained in \(R\), false otherwise.

R eq T: AlgMat, AlgMat -> BoolElt

Given a matrix algebra \(R\), and a matrix algebra \(T\), return true if \(R\) is equal to \(T\), false otherwise.

R ne T: AlgMat, AlgMat -> BoolElt

Given a matrix algebra \(R\) and a matrix algebra \(T\), return true if \(R\) is not equal to \(T\), false otherwise.