Anixx said:
Are there other better solutions?
Let me explain it differently.
There is only one way to understand the word
determinant over (for simplicity) a field ##\mathbb{F}##. Namely the group homomorphism
$$
\operatorname{det}\, : \,\operatorname{GL}(n,\mathbb{F}) \longrightarrow \mathbb{F}-\{0\}
$$
from the group of regular matrices over ##\mathbb{F}## to the group of units of ##\mathbb{F}##. It extents to all square matrices over ##\mathbb{F}## by adding ##\operatorname{det}(M)=0## for singular matrices.
There are at least four ways to understand
modulus.
- The ideal ##n\mathbb{Z}## in ##\mathbb{Z}## is called modulus in modular arithmetic, i.e. in all calculations in ##\mathbb{Z}_n.##
- A modulus is a formal product ##\prod_p p^{\nu(p)}\, , \,\nu(p)\geq 0## in algebraic number theory.
- A modulus is the absolute value of a real number.
- A modulus is the absolute value of a complex number.
The only possibility to compare these two terms is the possibility ##1##, since it is a group homomorphism, too, in that case. It is even a ring homomorphism. However, whereas we have all kind of fields in the case of a determinant, we have only discrete rings in case of a modulus. This is an important difference. Finite ideals cannot be summarized under infinite groups and vice versa. They are simply two completely different things.
Now you come along and suggest the following: Forget about the uniquely defined term
determinant and call it by the term
modulus which already has four meanings. Let's give it a fifth.
Why don't we call all and everything a modulus then? Would make life much easier. Or maybe not.
