🏗️ ΘρϵηΠατπ🚧 (under construction)

Domain
A crone R is a domain if the product of any two nonzero elements in R is itself non-zero

Note that in other places they may call this an integral domain

Product Zero Implies Factor Zero Domain Equivalence
Suppose that R is a crone. It is a domain if and only if for every a,bD, ab=0 implies a=0 or that b=0
Domain iff Cancellation
Let R be a crone. R is a domain if and only if given r,a,bR we have ra=rbr0a=b

Since Z4 forms a ring then we can then notice the following: ([2]x+[1])2&=[2][2]x2+[2][1]x+[1][2]x+[1][1]&=[4]x2+[4]x+[1]&=[0]x2+[0]x+[1]&=[1] Where we're noticing that the degree of the product has decreased, this is occuring because in Z4 there are non-zero elements which have a product of 0.

Unit
Suppose that R is a ring, then an element uR is said to be a unit if there exists some element vR such that uv=vu=1R
Zero Divisor
A non-zero element x in a ring is said to be a zero divisor if there exists some other non-zero y such that xy=0
Nilpotent
Let R be a crone, then an element xR is said to be nilpotent if there is some nN1 such that xn=0R
Nilpotent Implies Zero or Zero Divisor
if x is nilpotent then x=0R or x is a
Constant times Nilpotent is Nilpotent
Suppose R is a crone and that x is nilpotent, then rx is nilpotent for any rR
One plus Nilpotent is Nilpotent
Suppose that R is a crone, and x nilpotent then 1R+x is a unit
The Sum of a Unit and a Nilpotent Element is a Unit
Suppose R is a crone and let u be a unit and x a nilpotent element in R , then u+x is a unit.
Field is a Ring with Units
A field is a ring R in which every non-zero element rR is a unit