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That depends on what structure you want to put on either the topological or the algebraic side. Supposing your ring is a commutative $C^*$ algebra you can apply Gelfand duality to construct a compact hausdorff space. In this case the ring of functions are the continuous one. If you have a real f.g. algebra, you can construct an affine scheme over $\mathbb R$. The ring of functions would be polynomial functions. That is all part of the bigger picture of the duality between spaces and algebras ncatlab.org/nlab/show/Isbell+duality
@ToddTrimble : That's pretty interesting. Are you aware of any toposes in which the posets of ideals of a ring is not wellfounded? As to your question of a constructive proof, there's Jon Tennenbaums dissertation. Sadly, I can't access it.
You are right. It does not follow immediatly. I wrote that thinking an ideal in $R[X]$ is also an ideal in $R[[X]]$ under the inclusion, but this is wrong.
The standard definition of a strict $n$-category is as an $(n-1)$Cat-enriched category, like you said. However, there isn't a standard definition for weak $n$-categories. Up to tetra-categories (weak $4$-categories) there is an explicit algebraic definition; above that there are several different approaches, but as far as my knowledge goes, it is still unclear as to whether all of these are equivalent. I believe you mean the Trimble/May approach with "categorical objects in the category of categories"? ncatlab.org/nlab/show/algebraic+definition+of+higher+categories
Yes, I always meant weak higher categories, and in the same vein weak $\infty$-categories. The periodic table is less interesting when only looking at strict n-categories.
