Correct.phyguy321 said:If I is a subset of J then does that mean that f is in J
Sort of. I would just use the fact that J is generated by g(x).also and by definition of an ideal g*some b in J must equal something in J so g|f? because g|f means that f=bg for some b in J
An ideal is a subset of a ring that satisfies certain properties. It is a generalization of the concept of a multiple of a number in arithmetic. In a ring, an ideal is a subset that is closed under addition and multiplication by elements of the ring.
Ideals play a crucial role in abstract algebra and commutative algebra. They are used to study the structure and properties of rings, including their factorization and divisibility. Ideals also have applications in number theory, algebraic geometry, and coding theory.
An ideal is a special type of subset of a ring. It is a subset that contains the zero element of the ring and is closed under addition and multiplication by elements of the ring. This means that if any element of the ring is multiplied by an element in the ideal, the result will also be in the ideal.
A principal ideal is generated by a single element of a ring, while a non-principal ideal is generated by multiple elements. In a principal ideal, every element can be written as a multiple of the generator, while in a non-principal ideal, this may not be possible.
In a ring, an ideal can be thought of as a generalization of the concept of divisibility. If an element of a ring belongs to an ideal, it can be divided by any element in the ideal without leaving the ring. This is similar to how a number is divisible by its multiples in arithmetic.