The prime spectrum of a commutative ring is the underlying set of prime ideals of the ring together with the Zariski topology. A theorem proven by Reyes states that any extension of the set-valued prime spectrum functor on the category of commutative rings to the category of (not necessarily commutative) rings must assign the empty set to the n by n matrix algebra with complex entries when n is greater than 2. This suggests that sets do not serve as the underlying structure of a spectrum of a noncommutative ring. It is argued in his recent paper that coalgebras can be viewed as the underlying object of a noncommutative spectrum.

In this talk, we introduce coalgebras equipped with a ringed-space-like structure, which we call ringed coalgebras. These objects arise from fully residually finite-dimensional (RFD) algebras and schemes locally of finite type over a field k. The construction uses the Heyneman--Sweedler finite dual coalgebra and the Takeuchi underlying coalgebra. We will discuss that, if k is algebraically closed, the formation of ringed coalgebras gives a fully faithful functor out of the category of fully RFD algebras, as well as a fully faithful functor out of the category of schemes locally of finite type. The restrictions of these two functors to the category of (commutative) finitely generated algebras are isomorphic. In this way, ringed coalgebras can be thought of as a generalization of RFD algebras and schemes locally of finite type.