Tensor products of semirings (Q1345730)

By a semiring the author means an algebra \((S, 0, +, \cdot)\), where \((S, 0, +)\) is a commutative monoid with the identity 0, \((S, 0, \cdot)\) is a semigroup with the zero 0 and the multiplication ``\(\cdot\)'' distributes over the addition ``+'' from either side. Let \(\mathcal C\) be the category of semirings and semiring homomorphisms between them. For any semiring \(B\) there is the covariant functor \(- \otimes B : \mathcal C \to \mathcal C\), defined by \((- \otimes B) (A_ 1 @>f>> A_ 2) = A_ 1 \otimes B @>h>> A_ 2 \otimes B\), where \(\otimes\) is the tensor product of semirings and \(h(a_ 1 \otimes b) = f(a_ 1) \otimes b\) for all \(a_ 1 \in A_ 1\) and \(b \in B\). Similarly we can define the covariant functor \(B \otimes - \). The main results of this paper are to show that the tensor functors \(- \otimes B\) and \(B \otimes -\) preserve the exactness.