Operations on Elements of an Algebra

Operations on Elements

a + b: AlgGenElt, AlgGenElt -> AlgGenElt: The sum of the elements \(a\) and \(b\) of an algebra \(A\).

- a: AlgGenElt -> AlgGenElt: The negation of the algebra element \(a\).

a - b: AlgGenElt, AlgGenElt -> AlgGenElt: The difference of the elements \(a\) and \(b\) of an algebra \(A\).

a * b: AlgGenElt, AlgGenElt -> AlgGenElt: The product of the elements \(a\) and \(b\) of an algebra \(A\).

a * r: AlgGenElt, RngElt -> AlgGenElt
r * a: RngElt, AlgGenElt -> AlgGenElt: The product of the element \(a\) of the algebra \(A\) and the ring element \(r \in R\), where \(R\) is the coefficient ring of \(A\).

a / r: AlgGenElt, RngElt -> AlgGenElt: The product of the element \(a\) of the \(R\)-algebra \(A\) and the ring element \(1 / r \in R\), where \(R\) is a field.

a ^ n: AlgGenElt, RngIntElt -> AlgGenElt: If \(n\) is positive, form the (left-normed) \(n\)-th power \((( \ldots ((a*a)*a) \ldots )*a)\) of \(a\); if the parent algebra \(A\) of \(a\) has an identity and \(n\) is zero, return the identity; if \(n < 0\) and \(a\) has an inverse \(a^{-1}\) in \(A\), form the \(n\)-th power of \(a^{-1}\).

MinimalPolynomial(a): AlgGenElt -> RngUPolElt: If \(R\) is a field or the integer ring, return the minimal polynomial of the algebra element \(a\).

Parent(a): AlgGenElt -> AlgGen: For an element \(a\) in an algebra \(A\) return \(A\).

Comparisons and Membership

a eq b: AlgGenElt, AlgGenElt -> BoolElt: Returns true if the elements \(a\) and \(b\) of an algebra \(A\) are equal; otherwise false.

a ne b: AlgGenElt, AlgGenElt -> BoolElt: Returns true if the elements \(a\) and \(b\) of an algebra \(A\) are not equal; otherwise false.

a in A: AlgGenElt, AlgGen -> BoolElt: Returns true if \(a\) is in the algebra \(A\); otherwise false.

a notin A: AlgGenElt, AlgGen -> BoolElt: Returns true if \(a\) is not in the algebra \(A\); otherwise false.

Predicates on Elements

IsZero(a): AlgGenElt -> BoolElt: Returns true if the algebra element \(a\) is zero; otherwise false.

IsOne(a): AlgGenElt -> BoolElt: Returns true if the algebra element \(a\) is the identity; otherwise false.

IsMinusOne(a): AlgGenElt -> BoolElt: Returns true if the algebra element \(a\) is the negative of the identity element; otherwise false.

IsUnit(a): AlgGenElt -> BoolElt, AlgGenElt: Returns true if the element \(a\) is a unit, plus the inverse; otherwise false.

IsRegular(a): AlgGenElt -> BoolElt: Returns true if the element \(a\) is regular, that is, is not a zero divisor; otherwise false.

IsZeroDivisor(a): AlgGenElt -> BoolElt: Returns true if the algebra element \(a\) is a divisor of zero; otherwise false.

IsIdempotent(a): AlgGenElt -> BoolElt: Returns true if the element \(a\) is an idempotent, i.e. if \(a^2 = a\); otherwise false.

IsNilpotent(a): AlgGenElt -> BoolElt, RngIntElt: Returns true if the element \(a\) is nilpotent, i.e. if \(a^n = 0\) for some \(n \geq 0\); otherwise false. If true, the minimal \(n\) such that \(a^n = 0\) is returned as a second value.