K-theory of monoidal categories I am novice in the algebraic K- theory and don' t know if this is the right place for the following questions. So some people might consider them as basic questions.
Consider an exact monoidal category and its  K- groups as introduced by Quillen.
Do $K_n(\mathcal  C)$ have a ring structure? Clearly for n=0 the Grothendieck groups  have a ring structure induced by the tensor monoidal functor of $\mathcal C$. So bassically the question can be reformulated if the same tensor functor induces also a ring structure in the higher K - groups.
Next natural question would be if  a monoidal functor $ F: \mathcal C \rightarrow D$ induces a ring homomorphism at any level of the K- groups, $F^*_n:K_n(\mathcal C)\rightarrow K_n(\mathcal D)$?
I guess these are basic things but unfortunately I don't know any text where they are treated.
 A: (I'm not sure the term exact monoidal category is a standard one in the literature, so I'll just assume I know what you mean by it.)
Yes, $K_n$ commutes with products, as some have mentioned, but the tensor monoidal functor $\otimes : \mathcal C \times \mathcal C \to \mathcal C$ is not exact, so it doesn't induce a map $ K_n \mathcal C \times K_n \mathcal C \to K_n \mathcal C$.  Rather, the functor $\otimes : \mathcal C \times \mathcal C \to \mathcal C$ is bi-exact, in the sense that it is an exact functor in each of its two variables, separately.  (That's what makes $K_0 \mathcal C$ into a ring.)
The result is that the collection of abelian groups $ K_n \mathcal C$, indexed by $n$, forms a graded ring, with products $K_m \mathcal C \otimes K_n \mathcal C \to K_{m+n} \mathcal C$.  Assuming the tensor product is commutative up to natural isomorphism, the ring is commutative in the graded sense that $ x y = (-1)^{mn} y x$.
The actual construction is intricate, the main idea being to construct a map $K \mathcal C \wedge K \mathcal C \to K \mathcal C$ of spectra.  See, for example, Waldhausen's approach in section 9 of Algebraic K-theory of generalized free products. I, II. Ann. of Math. (2) 108 (1978), no. 1, 135–204.  It uses the Q-construction of Quillen, but could be simplified by using the S-construction of Segal that Waldhausen developed extensively.
My paper with Gillet, The loop space of the Q-construction, Illinois Journal of Mathematics, 31 (1987) 574-597, gives another approach that avoids both spectra (i.e., delooping) and the phony multiplication trap that Steve mentioned.
PS: My paper Algebraic K-theory via binary complexes, Journal of the American Mathematical Society, 25 (2012) 1149-1167, gives an algebraic description of the elements of the higher K-groups that converts the construction of the product into a simple exercise involving tensor product of chain complexes.
A: First, see Thomason's paper "Beware the Phony Multiplication on Quillen's $S^{-1}S$" (Proceedings of the AMS, 1990) for why the most obvious construction is wrong.  In the bibliography, you'll find references to papers of Loday, Waldhausen, Segal/Wolfson and May with constructions that are more complicated, but correct.
