Operations on Elements of an Algebra

Contents

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 ∈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 ∈R, where R is a field.
a ^ n : AlgGenElt, RngIntElt -> AlgGenElt
If n is positive, form the (left-normed) n-th power (( ... ((a * a) * a) ... ) * 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 a2 = a; otherwise false.
IsNilpotent(a) : AlgGenElt -> BoolElt, RngIntElt
Returns true if the element a is nilpotent, i.e. if an = 0 for some n ≥0; otherwise false. If true, the minimal n such that an = 0 is returned as a second value.
V2.28, 13 July 2023