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 a_1, ..., a_r of R, create the ideal I of R generated by a_1, ..., a_r.
Given a ring R and elements a_1, ..., a_r of R, construct the quotient ring Q = R/I, where I is the ideal of R generated by a_1, ..., a_r.
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 sum of the ideals I and J of the ring R. This ideal consists of elements a + b, with a in I and b in J. If I is generated by {a_1, ..., a_k} and J is generated by {b_1, ..., b_m}, then I + J is generated by {a_1, ..., a_k, b_1, ..., b_m}.
The product of the ideals I and J of the ring R. This is the ideal generated by elements a.b, with a in I and b in J, and it consists of elements a_1b_1 + ... + a_nb_n, with a_i in I and b_j in 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.
a in I : RngElt, RngIdl -> BoolElt
True if and only if the element a is a member of the ideal I.
True if and only if the element a is not a member of the ideal I.
True if and only if the ideals I and J are equal.
True if and only if the ideals I and J are distinct.
True if and only if the ideal I is contained in the ideal J.
True if and only if the ideal I is not contained in the ideal J.[Next] [Prev] [Right] [Left] [Up] [Index] [Root]