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Ideals and Quotient Rings

The following entries describe the operations on ideals in a commutative ring R. Certain operations on left and right ideals in non-commutative rings will be described in the online help nodes for the corresponding rings.

Subsections

Defining Ideals and Quotient Rings

ideal< R | a1, ..., ar > : Rng, RngElt, ..., RngElt -> RngIdl
Given a ring R and elements a1, ..., ar of R, create the ideal I of R generated by a1, ..., ar.
quo< R | ar, ..., ar > : Rng, RngElt, ..., RngElt -> Rng
Given a ring R and elements a1, ..., ar of R, construct the quotient ring Q = R/I, where I is the ideal of R generated by a1, ..., ar.
R / I : Rng, RngIdl -> Rng
Given a ring R and an ideal I of R, construct the quotient ring Q = R/I, as well as the canonical map R -> R/I.
PowerIdeal(R) : Rng -> PowIdl
The set of ideals of R. This is the parent of all ideals of R.

Arithmetic Operations on Ideals

I + J : RngIdl, RngIdl -> RngIdl
The sum of the ideals I and J of the ring R. This ideal consists of elements a + b, with a∈I and b∈J. If I is generated by {a1, ..., ak} and J is generated by {b1, ..., bm}, then I + J is generated by {a1, ..., ak, b1, ..., bm}.
I * J : RngIdl, RngIdl -> RngIdl
The product of the ideals I and J of the ring R. This is the ideal generated by elements a.b, with a∈I and b∈J, and it consists of elements a1b1 + ... + anbn, with ai∈I and bj∈J.
I meet J : RngIdl, RngIdl -> RngIdl
The intersection of the ideals I and J of the ring R.

Boolean Operators on Ideals

Throughout this subsection I and J are ideals belonging to the same integer ring R, while a is an element of R.

a in I : RngElt, RngIdl -> BoolElt
Returns true if and only if the element a is a member of the ideal I.
a notin I : RngElt, RngIdl -> BoolElt
Returns true if and only if the element a is not a member of the ideal I.
I eq J : RngIdl, RngIdl -> BoolElt
Returns true if and only if the ideals I and J are equal.
I ne J : RngIdl, RngIdl -> BoolElt
Returns true if and only if the ideals I and J are distinct.
I subset J : RngIdl, RngIdl -> BoolElt
Returns true if and only if the ideal I is contained in the ideal J.
I notsubset J : RngIdl, RngIdl -> BoolElt
Returns true if and only if the ideal I is not contained in the ideal J.
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Version: V2.19 of Mon Dec 17 14:40:36 EST 2012