# Basic question on the category CRing

abstract-algebracategory-theorycommutative-algebraring-theory

Let $$A$$ be a (commutative unitary) ring, let $$I\subseteq A$$ be a non-zero ideal. Are there any injective homomorphisms $$A\to A/I$$?

I've been thinking for a while, keeping in mind that the homomrphisms $$A/I\to R$$ are in natural bijection with the homomorpisms $$A\to R$$ whose kernels contain $$I$$ (for any ring $$R$$), but I couldn't prove anything. On the other hand I tried to find some examples, but without success. Can you give me any suggestion?

Sure. The trivial example is letting $$I = (0)$$ and taking the quotient map. But there are also plenty of examples with $$I \neq (0)$$.
The first example to come to me would be the following: Let $$A = k[x_1, x_2, \cdots]$$ be the polynomial ring with infinitely many variables and let $$I = (x_1)$$, so $$A/I \cong k[x_2, x_3, \cdots]$$. The map $$k[x_1, x_2, \cdots] \to k[x_2, x_3, \cdots], \ x_i \mapsto x_{i+1}$$ is an isomorphism (and in particular injective).