Whether we can define the finitely generated coideal?

Question

In the question How to understand the coidea of a colagebra?, we had posed the definition of coideals.

Let $R$ be a unitary commutative ring and $X$ be a finite subset of $R$. Then the ideal generated by $X$ can be expressed by $$\{r_1x_1s_1+r_2x_2s_2+\cdots+r_nx_ns_n\mid r_i,s_i\in r, x_i\in X\}.$$

My question is whether we can define the finitely generated coideal in the similar way? Is it well defined?

Answer 1

No we cannot (up to my knowledge of course).

However, there is something quite interesting happening here which might have some value for the OP. It is the fact that:

Although finitely generated coideals cannot be defined in a similar way, finitely generated subcoalgebras actually can, in the sense that they satisfy a somewhat dual "finiteness condition"; this is essentially the content of the so called fundamental theorem of coalgebras

Before i get into more details, it would be useful to recall the wider categorical setting (under the prism of duality) in which all these happen: Recall that algebras and coalgebras are dual objects (in finite dimensions this duality is very precise and simple) and under this duality, the subobjects correspond to factor objects and vice versa. Thus, the coideals of a coalgebra correspond to subalgebras of the dual algebra and the subcoalgebras of a coalgebra correspond to ideals of the dual algebra. (expressing these corespondences explicitly makes some really great exercices).

So, it is natural -from this point of view- to expect that a similar result (to the one described in the OP for ideals of algebras) should actually describe how a subobject (that is: a subcoalgebra) is generated rather than how a coideal is generated. As i mentioned earlier, this is actually the content of the so called "fundamental theorem of coalgebras" which (in one of its formulations) says that:

Every element $c\in C$ of a $k$-coalgebra $C$ is contained in a finite dimensional subcoalgebra.

Let me briefly desrcibe how this subcoalgebra (containing $c$) is generated:

Let $c\in C$ and $$(\Delta\otimes Id)\circ\Delta(c)=\sum_{i,j}c_i\otimes \alpha_{i,j}\otimes d_j$$ Now, take the $k$-subspace $A$ of $C$, generated (as a subspace) by the $a_{i,j}$. There is a finite number of them, thus $A$ has finite $k$-dimension. Now it can be shown that $c\in A$ and that $\Delta(A)\subseteq A\otimes A$, so $A$ is a finite dimensional subcoalgebra of $C$.
It is called: the subcoalgebra $A\subseteq C$ generated by $c\in C$, and it can be shown to be the "smallest" subcoalgebra of $C$ containing $c\in C$.

Some remarks:

1. The above argument can be easily extended to any finite subset $\{c_i\}$ of elements of $C$.
2. The fundamental theorem of coalgebras implies that: "Every coalgebra is the sum of its finite-dimensional subcoalgebras" (this is frequently referred to as the fundamental theorem). At least to my knowledge, there is no similar result for the case of algebras.
3. If you want some more details, proofs of the fundamental theorem of coalgebras can be found in most sources on Hopf algebras.