I'm new to ring in Abstract Algebra, and I just came across a theorem related to quotient ring and it kind of confused me.
The theorem: Let I be an ideal of R. The map φ: R → R/I defined by φ(r) = r + I is a ring homomorphism of R onto R/I with kernel I.
Not sure how to interpret this theorem.
The first way that I interpreted it:
Does this mean that we can only form a quotient ring if we have an Ideal with kernel? Which means that R/I exists only if it is kernel I.
Second interpretation:
It means that if we define a map φ: R → R/I and we want it to be a ring homomorphism, we need to have an Ideal with kernel?
Sorry for the question, but I don't know anybody that can answer my question.
Thank you,
Best Answer
The way this theorem is interpreted is that, for any ideal $I$ of the ring $R$, you can form a homomorphism from $R$ to $R/I$ by sending an element $x\in R$ to the coset $x+I$ in $R/I$. The kernel of the homomorphism is $I$.
Neither of your remarks are really related to the theorem, although they do have some meaning. First of all, notice that a quotient ring $R/I$ is defined by $I$ being an ideal. So for any ideal $I$ of $R$, you can form a quotient ring, and that construction/definition has nothing to do with this theorem. (You would need the definition of $R/I$ before stating this theorem). A quotient ring a priori has nothing to do with a homomorphism or a kernel, so your statement "$R/I$ exists iff $I$ is kernel" has no meaning.
Another thing to note is that the kernel of a homomorphism is always an ideal, which is a good exercise. However, you can have a homomorphism $R\to R/I$ that doesn't have a kernel of $I$.