An ideal in abstract algebra is a subset of a ring that satisfies certain properties. Specifically, it is a subset that is closed under addition and multiplication by elements of the ring. Ideals can also be thought of as generalizations of the concept of a normal subgroup in group theory.
A principal ideal is generated by a single element of the ring, while a non-principal ideal is generated by two or more elements. This means that a principal ideal has only one generator, while a non-principal ideal has multiple generators.
In abstract algebra, a generator is an element or subset of a mathematical structure that, when combined with other elements or operations, can create all other elements in that structure. For example, in a group, a single element can generate the entire group through repeated operations.
Yes, a ring can have multiple ideals. In fact, every ring has at least two ideals: the zero ideal, which contains only the additive identity element, and the ring itself. However, some rings may have infinitely many ideals.
Ideals and subrings are closely related concepts in abstract algebra. An ideal is essentially a subring that is closed under multiplication by elements of the ring. In other words, every ideal is a subring, but not every subring is an ideal.