Rings and abstract rings

[ghost]

In section 8 of the textbook, I defined rings as an abstract abelian group $(R, 0:R, +:R \times R \to R, -:R \to R)$ which additionally has the structure of elements $1:R$ and functions $(-)\cdot(-):R \times R \to R$ such that $(-)\cdot(-)$ distributes over $+$ and $1$ and $(-)\cdot(-)$ forms an associative H-space. However, on any pointed type $(A, a:A)$, any binary function $f:A \times A \to A$ induces a binary function on the loop space of $a$ by the binary action on paths of $f$, $\mathrm{ap}(f, a, a):\Omega(A, a) \times \Omega(A, a) \to \Omega(A, a)$.

This means that I could make the distinction between rings and abstract rings as the textbook does for groups and abstract groups in chapter 4, and I could define an abstract ring as above, and a ring as an abelian group $(R, \mathrm{sh}_R:B R, z_R:Z(R) \simeq R)$ with a loop $l:\Omega(B R, r)$ and a function $\mu:B R \times B R \to B R$ such that $\mathrm{ap}(\mu, \mathrm{sh}_R, \mathrm{sh}_R)$ distributes over concatenation of paths in the loop space $\Omega(B R, \mathrm{sh}_R)$, and $(\Omega(B R, \mathrm{sh}_R), l, \mathrm{ap}(\mu, \mathrm{sh}_R, \mathrm{sh}_R))$ forms an associative H-space.

In fact, I could get rid of the requirement that group be abelian, because given a group $R$, if there is an H-space structure $(\Omega(B R, \mathrm{sh}_R), l, \mathrm{ap}(\mu))$ on the loop space of the designated shape $\mathrm{sh}_R:B R$ such that $\mathrm{ap}(\mu)$ distributes over concatenation of paths in the loop space $\Omega(B R, \mathrm{sh}_R)$, then the group is abelian, because for all paths $p:\Omega(B R, \mathrm{sh}_R)$ and $q:\Omega(B R, \mathrm{sh}_R)$, there are paths
$$p * p * q * q = \mathrm{ap}(\mu, \mathrm{sh}_R, \mathrm{sh}_R)(p, l * l) * \mathrm{ap}(\mu, \mathrm{sh}_R, \mathrm{sh}_R)(q, l * l) = \mathrm{ap}(\mu, \mathrm{sh}_R, \mathrm{sh}_R)(p * q, l * l)$$
$$\mathrm{ap}(\mu, \mathrm{sh}_R, \mathrm{sh}_R)(p * q, l * l)= \mathrm{ap}(\mu, \mathrm{sh}_R, \mathrm{sh}_R)(p * q, l) * \mathrm{ap}(\mu, \mathrm{sh}_R, \mathrm{sh}_R)(p * q, l) = p * q * p * q$$
and by the action on paths of the function $\lambda r:\Omega(B R, \mathrm{sh}_R).p^{-1} * r * q^{-1}$, one gets the path $p * q = q * p$, which implies that the group is abelian.

The results for abstract rings should follow from taking the underlying set $U(R)$ of a ring.