In physics we deal with quantities which have a magnitude and a unit type, such as 4m, 9.8 m/s², and so forth. We might represent these as elements of $\Bbb R\times \Bbb Q^n$ (where there are $n$ different fundamental units), with $(4, \langle1,0,0\rangle)$ representing 4m, $(9.8, \langle1,0,-2\rangle)$ representing 9.8 m s⁻², and so forth, with the following rules for arithmetic:

- $(m_1, u_1) + (m_2, u_2)$ is defined if and only if $u_1 = u_2$, in which case the sum is $(m_1 + m_2, u_1)$.
- $(m_1, u_1) \cdot (m_2, u_2)$ is always defined, and is equal to $(m_1m_2, u_1+u_2)$, where $u_1+u_2$ is componentwise addition of rationals.

This structure has a multiplicative identity, namely $(1, 0)$, and a family of additive quasi-identities, namely $(0, u)$ for each $u$. We have $$(1, 0)\cdot (m, u) = (m,u)\cdot(1,0) = (m, u)$$ and $$(0, u) + (m, u) = (m,u)+(0,u) = (m, u)$$ for every $(m, u)$, but in the latter case the quasi-identity $(0,u)$ isn't a constant; it depends on the $u$ part of $(m,u)$.

Every element with $m\ne 0$ has a multiplicative inverse, and every element $(m, u)$ has an additive quasi-inverse $(-m, u)$ with $(m,u) + (-m, u) = (0, u)$, where $(0,u)$ is an additive quasi-identity. Multiplication distributes over addition. If either side of $$p\cdot(q+r) = (p\cdot q)+ (p\cdot r)$$ is defined, then so is the other, and they are equal.

All taken together this is very much like a field, except that the additive identity is peculiar. There is a $0_\text{m}$, a $0_\text{s}$ and a $0_\text{kg}$, represented as $(0, \langle1,0,0\rangle), (0, \langle0,1,0\rangle), $and $ (0, \langle0,0,1\rangle)$, and they can be multiplied but not added.

We might generalize this slightly, and define the same sort of structure over a ring $\langle G,+, \cdot\rangle$ and a group $\langle H,\star\rangle$: $G❄H$ is an algebraic structure whose elements are elements of $G\times H$, where $(g_1, h_1) + (g_2, h_2)$ is defined to be $(g_1+g_2, h_1)$ if and only if $h_1 = h_2$, and $(g_1, h_1) \times (g_2, h_2)$ is defined to be $(g_1\cdot g_2, h_1\star h_2)$ always. (Or we might relax the condition on $H$ and make it a monoid, or whatever.)

Does this thing have a name? Is it of any interest? Are there any interesting examples other than the one I started with?

I did observe that this structure is also a bit like floating-point numbers, where the left component is the mantissa and the right component the exponent, except that floating-point numbers also have a normalization homomorphism that allows one to add $(m_1, e_1)$ and $(m_2, e_2)$ even when $e_1\ne e_2$, and to understand $(m_1, e_1)$ and $(b\cdot m_1, e_1 - 1)$ as different representations of the same thing.