A normed ring "should" be a monoid object in the monoidal category of normed abelian groups. There are (at least) two choices of morphisms of normed groups, namely bounded or short homomorphisms, resulting in two definitions of a normed ring: Concretely, a normed ring is a ring $R$ (not necessarily commutative) together with a map $R_{\mathsf{Set}} \to \mathbb{R}_{\geq 0},~ x \mapsto |x|$ such that for all $x,y \in R_{\mathsf{Set}}$

- $|x+y| \leq |x|+|y|$
- $|{-}x|=|x|$
- $|x|=0 \Leftrightarrow x=0$,

and now either (choosing short homomorphisms)

- $|1| \leq 1$ and $|x y| \leq |x| |y|$ for all $x,y$,

(which easily implies $R = 0$ or $|1|=1$), equivalently

- $|x_1 \cdot \dotsc \cdot x_n| \leq |x_1| \cdot \dotsc \cdot |x_n|$ for every finite sequence of elements $x_1,\dotsc,x_n$ (including the empty one!),

or (choosing bounded homomorphisms)

- there is a constant $K \geq 0$ with $|xy| \leq K|x||y|$ for all $x,y$

(but no condition for the unit). In all the references that I have found so far, these two definitions are somewhat mixed. For example, $|xy| \leq |x||y|$ is assumed, but nothing about the unit. Isn't this a mistake? Also, I have almost never found the condition $|{-}x|=|x|$, but this doesn't seem to follow from the rest, right? What do you think about the stronger multiplicativity condition $|xy|=|x||y|$? This prevents zero divisors, which is certainly useful, but should one really put this into the definition? Also, which of the two definitions (short/bounded) is preferred in which contexts, and why? I would like to ask similar questions for normed modules over a normed ring, but maybe this will be a separated post.