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.
Given a ring R and elements a1, ..., ar of R, create the ideal I of R generated by a1, ..., ar.
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.
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.
The set of ideals of R. This is the parent of all ideals of R.
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}.
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.
The intersection of the ideals I and J of the ring R.
Throughout this subsection I and J are ideals belonging to the same integer ring R, while a is an element of R.
Returns true if and only if the element a is a member of the ideal I.
Returns true if and only if the element a is not a member of the ideal I.
Returns true if and only if the ideals I and J are equal.
Returns true if and only if the ideals I and J are distinct.
Returns true if and only if the ideal I is contained in the ideal J.
Returns true if and only if the ideal I is not contained in the ideal J.[Next][Prev] [Right] [Left] [Up] [Index] [Root]