On the dimension of the space of integrals on coalgebras

We study the injective envelopes of the simple right $C$-comodules, and their duals, where $C$ is a coalgebra. This is used to give a short proof and to extend a result of Iovanov on the dimension of the space of integrals on coalgebras. We show that if $C$ is right co-Frobenius, then the dimension of the space of left $M$-integrals on $C$ is $\leq {\rm dim}M$ for any left $C$-comodule $M$ of finite support, and the dimension of the space of right $N$-integrals on $C$ is $\geq {\rm dim}N$ for any right $C$-comodule $N$ of finite support. If $C$ is a coalgebra, it is discussed how far is the dual algebra $C^*$ from being semiperfect. Some examples of integrals are computed for incidence coalgebras.