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

Ring Homomorphism
Suppose that (R,โŠ•,โŠ—) and (S,โŠž,โŠ ) are rings, and ฯ•:Rโ†’S is a function, then ฯ• is a ring homomorphism if for any a,bโˆˆR
  • ฯ•(aโŠ•b)=ฯ•(a)โŠžฯ•(b)
  • ฯ•(aโŠ—b)=ฯ•(a)โŠ ฯ•(b)
Ring with Unity Homomorphism
Suppose that R,S are rings with unity, then we say that ฯ• is a homomorphism between R and S when ฯ• is a ring homomorphism and we have ฯ•(1R)=1S
Crone Homomorphism
Suppose that R,S are crones, then if ฯ• is a ring with unit homomorphism, then we say that ฯ• is a crone homomorphism

We may also say that ฯ• is a homomorphism from R to S, or that ฯ• is a homomorphism of rings

Crone Homomorphism Between Two by Two Matices with 0 in Bottom Left and Integers
Let R:={(ab0a):a,bโˆˆZ} along with the usual definition for matrix addition and matrix multiplication, and define ฯ•:Rโ†’Z be defined such that ฯ•((ab0a))=a Show the following
  • ฯ• is a crone homomorphism
  • Find ker(ฯ•)
  • Show R/ker(ฯ•) is ismorphic to Z
  • is ker(ฯ•) a prime ideal?
Ring Isomorphism
Suppose that ฯ• is a ring homomorphism and is a bijection, then we say that ฯ• is a ring isomorphism
Homomorphisms carry Zero to Zero
Suppose that ฯ•:Rโ†’S is a homomorphism of rings, then ฯ•(0R)=0S