# Ideal Operations

## `ideal< R | a₁, ..., aᵣ >: RngIntRes, RngIntResElt, ..., RngIntResElt -> RngIntRes`

The ideal of the residue ring $R$ generated by the greatest common divisor of the elements $a_i$ and the modulus of $R$.

## `GreatestCommonDivisor(a, b): RngIntResElt, RngIntResElt -> RngIntResElt`

## `Gcd(a, b): RngIntResElt, RngIntResElt -> RngIntResElt`

## `GCD(a, b): RngIntResElt, RngIntResElt -> RngIntResElt`

Greatest common divisor of the elements $a$ and $b$ of $R$, that is, a generator for the $R$-ideal $(a)+(b)$.

## `GreatestCommonDivisor(Q): [RngIntResElt] -> RngIntResElt`

## `Gcd(Q): [RngIntResElt] -> RngIntResElt`

## `GCD(Q): [RngIntResElt] -> RngIntResElt`

Greatest common divisor of the sequence of elements $Q$, that is, a generator for the $R$-ideal generated by the elements in $Q$.

## `LeastCommonMultiple(a, b): RngIntResElt, RngIntResElt -> RngIntResElt`

## `Lcm(a, b): RngIntResElt, RngIntResElt -> RngIntResElt`

## `LCM(a, b): RngIntResElt, RngIntResElt -> RngIntResElt`

Least common multiple of the elements $a$ and $b$ of $R$, that is, a generator for the $R$-ideal $(a)\cap (b)$.

## `LeastCommonMultiple(Q): [RngIntResElt] -> RngIntResElt`

## `Lcm(Q): [RngIntResElt] -> RngIntResElt`

## `LCM(Q): [RngIntResElt] -> RngIntResElt`

Least common multiple of the sequence of elements $Q$, that is, a generator for the $R$-ideal formed by the intersection of the principal ideals generated by elements of $Q$.

## `I + J: RngIntRes, RngIntRes -> RngIntRes`

## `I * J: RngIntRes, RngIntRes -> RngIntRes`

## `I meet J: RngIntRes, RngIntRes -> RngIntRes`

## `a in I: RngIntResElt, RngIntRes -> BoolElt`

## `a notin I: RngIntResElt, RngIntRes -> BoolElt`

## `I eq J: RngIntRes, RngIntRes -> BoolElt`

## `I ne J: RngIntRes, RngIntRes -> BoolElt`

## `I subset J: RngIntRes, RngIntRes -> BoolElt`

## `I notsubset J: RngIntRes, RngIntRes -> BoolElt`