An ideal of a commutative ring is a subset of the ring that is closed under addition, subtraction, and multiplication by elements of the ring. It is also absorbent, meaning that if an element of the ideal is multiplied by any element of the ring, the result is still in the ideal.
To prove that a subset is an ideal of a commutative ring, you must show that it satisfies the definition of an ideal. This includes proving that it is closed under addition and multiplication, and that it is absorbent. You also need to show that it is a subset of the ring and that it contains the additive identity element of the ring.
A proper ideal is a subset of a commutative ring that is also an ideal, but does not contain all elements of the ring. An improper ideal is a subset of a commutative ring that is also an ideal and contains all elements of the ring. In other words, a proper ideal is a strict subset of the ring, while an improper ideal is the entire ring.
Yes, a commutative ring can have multiple ideals. In fact, every commutative ring has at least two ideals - the trivial ideal, which only contains the additive identity element, and the entire ring itself. Other commutative rings may have an infinite number of ideals.
Ideals in a commutative ring are analogous to normal subgroups in group theory. Just as an ideal is a subset of a ring that is closed under addition, subtraction, and multiplication, a normal subgroup is a subset of a group that is closed under multiplication and inverse operations. Additionally, both ideals and normal subgroups are used to define quotient structures.