Are algebraic properties consistent among ALL types of number groups? I'm embarking on a self study course of group theory, modular arithmetic, and other mathematics relating to cryptography.
I notice that when studying modular arithmetic that they explicitly say that the relation is 

compatible with the operations of the ring of integers: addition, subtraction, and multiplication.

This implies that there are times that that these operations may not operate as expected.  Should I proceed with my understanding that the fundamental aspects of mathematics will not apply to different groups of numbers (in a ring (whatever that is)) or in any other aspect of math?
I feel like I need to question my fundamental understanding of math, primarily because I am so overwhelmed with foreign symbols and terminology that I'm just trying to prevent even more misunderstandings.
 A: You can define structures with whatever kind of operations or relations you want to, as long as you do so in a way that is self-consistent (leads to no contradictions).
There are structures that have addition, subtraction, multiplication and division with both additive and multiplicative identities in which everything is commutative, associative and distributive (i.e. a field), there are structures that have all these things but perhaps commutativity of multiplication (division algebras), or without division too (unital rings), without a multiplicative identity (non-unital rings), without subtraction (semirings), or even without associativity (e.g. Lie algebras, but those have a different type of structure, involving the Jacobi identity).
There are structures that have only one binary operation which is associative and has an identity and inverses (groups), or without inverses (monoids), or without identities (semigroups), or even without associativity (magmas), or partially defined operation (e.g. groupoid).
There are structures with special relations between elements, like a partially ordered set. These include all types of posets and lattices, possibly with additional structure (operations which preserve all relations when applied to the things appearing within them, e.g. $x>y\Rightarrow x+1>y+1$).
There are structures with special subsets that dictate how "close" elements are to each other, called open sets, which themselves have a number of properties that they collectively satisfy (closure under finite intersections and arbitrary unions), which in turn then distinguish special types of maps (called continuous maps) and distinguish a heirarchy of topological properties (for example compactness, connectedness, and a battery of separation axioms).
Modular arithmetic is a specific accessible example of a quotient structure. For modular arithmetic, if $a\equiv b$ mod $n$ and $c\equiv d$ mod $n$, then $a+c\equiv b+d$ and $ac\equiv bd$ mod $n$. That is, performing the operations of addition and multiplication preserve the relation of being congruent modulo a given number $n$. This motivates more general forms of quotients, namely of rings and groups.
Relations that are preserved by operations are generally interesting and worthy of study (and occasionally are of fundamental importance in defining some things, like knots or fundamental groups), but it is easy to make up relations that are not preserved by operations, and often there are "naturally ocurring" relations which are not preserved by operations.
Ultimately, we make the rules. We make up a set of rules for any given structure. Then we follow those rules and see what facts we can derive from those sets of rules using logical reasoning.
There are many introductory texts and notes out there on the topics of:


*

*Group theory specifically, or

*Groups, rings and fields (intro abstract algebra), or

*Field theory and Galois theory (latter requiring groups), or

*Topological spaces (called "general topology"), etc.


There are also texts on Lie algebras, but this is a more advanced subject (especially because of the difficult of differential geometry). Order theory is just something I've brushed up against here and there, and I imagine specific discussions of ordered algebraic structures are more niche. Universal algebra and modern algebra at a more advanced general level will cover a lot more algebraic structures, like monoids, groupoids and whatnot. (These are not covered at an introductory level probably because of the legwork-to-get-results / beauty-of-first-results ratio.)
I am studying algebraic number theory right now. It uses a lot of ring theory / commutative algebra notions but in the particular setting of structures that are very numbery, for lack of a more direct and official description. In particular it studies number fields and structures associated to them (rings of integers, prime ideals factorizations, localizations and completions, Galois groups and residue fields, and so on). When I tell others what the math I study (currently) is about, I usually say I study "the structure of number systems."
The fact that entire branches of math have been devoted to different structures tells us there is a rather wide range of configurations of rules possible to impose on structures, and furthermore that the resulting "theory" (sets of facts and theorems and formulae we can derive from the rules) is very rich and, at least to the sensibilities of modern mathematicians, beautiful.
