Is there a name and a particular symbol for the operation $+,\cdot$ on $\mathbb{Z}_n$?

Question

The question is very short:

Is there a name and a particular symbol for the operation $+$ and $\cdot$ on $\mathbb{Z}_n$? Such as "module $6$ addition" or "$\cdot_{15}$".

I have searched some books, but don't get one. When talking about $\mathbb{Z}_n$(and I am currently writing some self notes on abstract algebra), if there's no such terminology for $+$ and $\cdot$, this would be so inconvienient. Also, when talking about $\mathbb{Z}_4$ and $\mathbb{Z}_5$ at the same time, there's no distinct symbols for $+$(or $\cdot$), which is also inconvenient.

Answer 1

Addition and multiplication modulo $n$.

Answer 2

Some textbooks use $\oplus$ and $\odot$ to distinguish the operations on $\mathbb{Z}_n$ from the operations on the real numbers. (In latex these are typset with \oplus and \odot.) But this is usually considered "teddy bear," which means that it's expected that the once the student gets the idea, he'll drop the notation and just use $+$ and $\cdot$ like everyone else.

If you have a particular need to distinguish $+$ in $\mathbb{Z}_4$ from $+$ in $\mathbb{Z}_5$, you could use $\oplus_4.$ (I think I've seen this somewhere.)

Answer 3

You may encounter names like modular addition or addition mod $n$ with (nonstandard) operation symbols visually close to the usual symbols but with additional annotation. But generally this is discouraged. Rather in abstract or universal algebra we strive to use uniform unversal notation for the operations of algebraic structures of the same type (here rings).

There are many good reasons for such uniformity. For example, it allows us to write universal identities for commutative rings such as the Binomial Theorem or $\,x^2-y^2 = (x-y)(x+y)\,$ in universal form, using uniform notation for the operations (such uniformity which makes it easier for us to recognize such patterns when they occur, e.g. Remark below has a nontrivial example).

Further, it allows us to match up corresponding operations in related structures without ambiguity (i.e. without having to guess from the "shape" of the operation symbols which is addition or multiplication, etc). This allows us to universally define concepts like homomorphisms $\,h(a+b) = h(a)+h(b)\,$ (note the same notation for '$+$' on both sides), and substructures, etc. For more on this, see this excerpt on "Naming of Parts" from W. Hodges Model Theory textbook.

This may seem like pendantry when one first begins studying abstract algebra, but the reasons will be clarified if one later studies general algebras and structures in universal algebra, esp. when one learns about the beautiful deep connections between syntax and semantics in model theory, e.g. Birkhoff's HSP Theorem.

Remark $ $ Below is said nontrivial example on gcds from this post. Here $\,(x,y)\,$ denotes $\,\gcd(x,y),\,$ and, like addition, gcd is associative and commutative and multiplication distributes over it, so we can treat it similarly to addition, denoting the gcd by $\dot +$ in infix notation, which clarifies proofs, e.g.

$\qquad\ \ \ \color{#c00}{ab = cd}\ \Rightarrow\ (a,\,c)\,(a,\,d) = (aa,\,\color{#c00}{cd},\,ac,\,ad)\, =\, \color{#c00}a\,(a,\color{#c00}b,c,d),\ $ or, $ $ in infix notation:

$\qquad\ \ \ \color{#c00}{ab = cd}\ \Rightarrow\, (a\dot+c)\,(a\dot+d) = aa\dot+\color{#c00}{cd}\dot+ac\dot+ad\, =\, \color{#c00}a(a\dot+\color{#c00}b\dot+c\dot+d)$

using basic GCD laws (distributive, commutative, associative). The proof is much more intuitive using this infix notation for the gcd since it allows us to exploit our well-honed intuition on patterns of integer arithmetic. One way to abstract out this common structure of gcd and integer arithmetic employs certain types of lattices. But I digress.