๐Ÿ—๏ธ ฮ˜ฯฯตฮทฮ ฮฑฯ„ฯ€๐Ÿšง (under construction)

Ideal
An ideal in a crone (R,โŠ•,โŠ—) is a subset I containing 0R such that
  • a,bโˆˆI implies that aโˆ’bโˆˆI
  • aโˆˆI and rโˆˆR implies that rโŠ—aโˆˆI
Trivial Ideal
Suppose R is a crone, then {0R} is an ideal in R
Every Crone is an Ideal
For any crone R it is an ideal in R
The Kernel of a Crone Homomorphism is a Proper Ideal
If ฯ•:Rโ†’S is a crone homomorphism then, then ker(ฯ•) is a proper ideal in R
A Crone Homomorphism is an Injection iff ker(ฯ•)={0R}
If ฯ•:Rโ†’S is a crone homomorphism then, ฯ• is injective iff ker={0R}
Proper Ideal
We say that an ideal I in a crone R is proper when Iโ‰ R
An Ideal is a Normal Subgroup
The Intersection of Ideals is an Ideal
Suppose that I is a family of ideals in a crone R then โ‹‚I is an ideal in R
Ideal Generated By a Set
Suppose that R is a crone and that XโŠ†R then we define the ideal generated by X as the intersection of all ideals in R that contain X, symbolically let IX be the family of all ideals that contain X, then (X)โ‹„:=โ‹‚IX
Ideal Lightened Notation
We define the notation (a1,a2,โ€ฆ,an)โ‹„:=({a1,a2,โ€ฆ,an})โ‹„ to lighten the notation.
Ideal Generated by a Set is an Ideal
(X)โ‹„ is an ideal in R
Ideal Generated by a Set is the Smallest Ideal Containing the Set
Suppose R is a crone and XโŠ†R , then XโŠ†(X)โ‹„ and for any other ideal J such that XโŠ†J we have (X)โ‹„โŠ†J
Left Multiplication Yields an Ideal
Suppose that aโˆˆR then RโŠ—{a} is an ideal
Principal Ideal Generated by An Element
Suppose that R is a crone and that aโˆˆR, then RโŠ—a is the principal ideal generated by a
Principal Ideal Equals Generated Ideal
RโŠ—{a}=(a)โ‹„
Ideal Generated by a Finite Set is their Linear Combinations
Let A={a1,a2,โ€ฆan} For any nโˆˆN1 we have (A)โ‹„={โˆ‘i=1nriai:riโˆˆR,iโˆˆ[1...n]}
Product of Ideals is Contained in their Intersection
Let I,J be two ideals and define their product IJ:={โˆ‘i=1nriaibi:aiโˆˆI,biโˆˆJ,riโˆˆR,nโˆˆN1} Prove that IJโŠ†IโˆฉJ
Can't get to All Polynomials From a Generated Ideal
Let I be the ideal in Z[x] generated by {2,x}, prove that I2:=II contains elements not of the form ab for a,bโˆˆI
An Ideal of Continuous Functions
Let R:=C([0,1]) be the set of continuous functions f:[0,1]โ†’R. For any cโˆˆR we define Ic:={fโˆˆR:f(c)=0}
  • Show that the set R is a crone, and the set Ic is an ideal of R
  • Is Ic1โˆชIc2 an ideal? What about Ic1โˆฉIc2?
  • Show that R/Icโ‰…R Hint: consider the map ฯ•(f+Ic)=f(c)
Quotient Remainder For Polynomials
Suppose that R is a domain, and that f(x),g(x)โˆˆR[x], then there exists unique polynomials q(x),r(x)โˆˆR[x] such that f(x)=q(x)g(x)+r(x) where either r(x)=0R or deg(r)โ‰คdeg(g)