Principal ideal

In the mathematical field of ring theory, a principal ideal is an ideal I {displaystyle I} in a ring R {displaystyle R} that is generated by a single element a {displaystyle a} of R {displaystyle R} through multiplication by every element of R {displaystyle R} . The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset P {displaystyle P} generated by a single element x ∈ P {displaystyle xin P} , which is to say the set of all elements less than or equal to x {displaystyle x} in P {displaystyle P} . In the mathematical field of ring theory, a principal ideal is an ideal I {displaystyle I} in a ring R {displaystyle R} that is generated by a single element a {displaystyle a} of R {displaystyle R} through multiplication by every element of R {displaystyle R} . The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset P {displaystyle P} generated by a single element x ∈ P {displaystyle xin P} , which is to say the set of all elements less than or equal to x {displaystyle x} in P {displaystyle P} .