Fix an $n\in\mathbb{N}$ and a field $\mathbb{K}$.
A lot of texts in linear algebra like to define the determinant function on $\operatorname{M}_n\left(\mathbb{K}\right)$ as the unique function $\operatorname{M}_n\left(\mathbb{K}\right) \to \mathbb{K}$ which is alternating and multilinear on the rows of the matrix and sends the identity matrix $I_n$ to $1$. This definition generalizes verbatim to the case when $\mathbb{K}$ is a commutative ring.
Jim Hefferon's Linear Algebra (version 22 Dec 2014) (Definition 2.1 in Chapter Four) uses a slightly modified version of this definition. In my notations, a Hefferonian determinant function means a function $f : \operatorname{M}_n\left(\mathbb{K}\right) \to \mathbb{K}$ having the following four properties:
If $A \in \operatorname{M}_n\left(\mathbb{K}\right)$ and $B \in \operatorname{M}_n\left(\mathbb{K}\right)$ are such that $B$ is obtained from $A$ by adding a multiple of a row of $A$ to another row of $A$, then $f\left(B\right) = f\left(A\right)$.
If $A \in \operatorname{M}_n\left(\mathbb{K}\right)$ and $B \in \operatorname{M}_n\left(\mathbb{K}\right)$ are such that $B$ is obtained from $A$ by switching two rows, then $f\left(B\right) = - f\left(A\right)$.
If $A \in \operatorname{M}_n\left(\mathbb{K}\right)$ and $B \in \operatorname{M}_n\left(\mathbb{K}\right)$ are such that $B$ is obtained from $A$ by multiplying a row by a scalar $\lambda$, then $f\left(B\right) = \lambda f\left(A\right)$.
We have $f\left(I_n\right) = 1$.
Hefferon then shows that such a function $f$ is unique when $\mathbb{K}$ is a field. It is easy to show that, more generally, there is a unique Hefferonian determinant function when $\mathbb{K}$ is an integral domain (namely, the usual determinant $\det$).
Out of curiosity, I am wondering how far this uniqueness statement can be generalized. It doesn't feel right to expect it to hold over an arbitrary commutative ring $\mathbb{K}$, as things like Gaussian elimination just will not work and there will not be any quotient field to salvage them. But I am not sure how to find a counterexample. Ideas?
Notice that Hefferon is not the only author who defines a determinant in such a strange way. The definition of a determinant in Theorem 1.50 of Peter J. Olver's and Chehrzad Shakiban's Applied Linear Algebra (2006) is similar to Hefferon's. Instead of property 4, it requires $f$ to send any upper-triangular matrix to the product of its diagonal entries. This is stronger than Hefferon's property 4, and I am wondering if it is actually stronger or just equivalent?
Apparently the popularity of these unnatural definitions is due to the opinion that genuine multilinearity is too difficult for students to grasp; I am not sure if this justifies them, but I believe that the question it posts is interesting!