Definition
For a morphism of rings , we define the image of ,
This is the function theoretic image. We also have that the image is a subring .
Remark
If is a morphism of unital rings, then implies that is a unital subring.
For a morphism of rings f:(R,+,⋅)→(S,+,⋅), we define the image of f,
imf=f(R)={f(r)∣r∈R}.This is the function theoretic image. We also have that the image is a subring imf≤S.
If f is a morphism of unital rings, then f(1R)=1S∈imf implies that imf is a unital subring.