of these two modules is cohopfian, R is hopfian on both sides (since R is projective as a left or right module) and directly finite. Hopfian and cohopfian rings The situation in the category of rings is quite different from the category of modules. The morphisms in the category of rings with unity are required to preserve the identity, that is, to send 1 to 1. If R satisfies the ascending chain condition on ideals, then R is hopfian. This can be proven by analogy with the fact for Noetherian modules. The counterpart idea for "cohopfian" does not exist however, since if f is a ring homomorphism from R into R preserving identity, and the image of f is not R, then the image is certainly not an ideal of R. In any case, this shows that a one sided Noetherian or Artinian ring is always hopfian. Any simple ring is hopfian, since the kernel of any endomorphism is an ideal, which is necessarily zero in a simple ring. In contrast, in , an example of a non-cohopfian field was given. The full linear ring EndD(V) of a countable dimensional vector space is a hopfian ring which is not hopfian as a module, since it only has three ideals, but it is not directly finite. The paper also gives an example of a cohopfian ring which is not cohopfian as a module. Also in , it is shown that for a Boolean ring R and its associated Stone space X, the ring R is hopfian in the category of rings if and only if X is cohopfian in the category of topological spaces, and R is cohopfian as a ring if and only if X is hopfian as a topological space. Hopfian and cohopfian topological spaces In , a series of results on compact manifolds are included. Firstly, the only compact manifolds which are hopfian are finite discrete spaces. Secondly, compact manifolds without boundary are always cohopfian. Lastly, compact manifolds with nonempty boundary are not cohopfian. References External links Hopfian group Co-hopfian group Category theory Group theory Module theory Ring theory