I am currently reading up on normed algebra and am a bit confused about the conventions regarding the multiplicative properties of a norm.
In a normed ring the norm is only supposed to be submultiplicative in the sense that $\vert xy \vert \leq \vert x \vert\vert y\vert$. This is important, since for example most matrix-norms are only submultiplicative, and so are the norms of most Banach-algebras.
On the other hand a normed vector space is assumed to have a multiplicative norm in the sense that $\Vert\lambda \cdot v\Vert = \vert \lambda \vert \cdot \Vert v\Vert$. I would assume this will be the same for normed modules.
The problem with these definitions is: a normed ring is not a normed module over itself. To be fair, I am not sure how big of a problem it is. Most normed rings we consider modules over (like $\mathbb{Z,R,C},\mathbb{Z}_p,\mathbb{Q}_p, \mathbb{C}_p$) do have multiplicative norms. And I have not yet seen modules over general Banach-algebras.
So what is the correct thing to do here? To only consider normed modules over multiplicatively normed rings or to generalize the notion of normed module to only require submultiplicativity?
The right definition is in fact $|xy|\leq|x||y|$, and this is the standard definition to use when talking about normed modules over normed rings. Note, however, that if $x$ is invertible and $|x^{-1}|=|x|^{-1}$, then we also have $|y|=|x^{-1}xy|\leq |x^{-1}||xy|=|x|^{-1}|xy|$ so $|xy|\geq |x||y|$ and thus $|xy|=|x||y|$. In particular, this is always true for the scalar multiplication on a normed vector space, if the norm on the field of scalars is multiplicative (which it is for the usual examples of $\mathbb{R},\mathbb{Q}_p$, etc. So for vector spaces (over a field with a multiplicative norm), submultiplicative norms and multiplicative norms are the same.