Ring Homomorphism
Suppose that $$\left( R, \oplus, \otimes \right)$$ and $$\left( S, \boxplus, \boxtimes \right)$$ are rings, and $$\phi: R \to S$$ is a function, then $$\phi$$ is a ring homomorphism if for any $$a, b \in R$$
• $$\phi \left( a \oplus b \right) = \phi \left( a \right) \boxplus \phi \left( b \right)$$
• $$\phi \left( a \otimes b \right) = \phi \left( a \right) \boxtimes \phi \left( b \right)$$
Ring with Unity Homomorphism
Suppose that $$R, S$$ are rings with unity, then we say that $$\phi$$ is a homomorphism between $$R$$ and $$S$$ when $$\phi$$ is a ring homomorphism and we have $$\phi \left( 1 _ R \right) = 1 _ S$$
Crone Homomorphism
Suppose that $$R, S$$ are crones, then if $$\phi$$ is a ring with unit homomorphism, then we say that $$\phi$$ is a crone homomorphism

We may also say that $$\phi$$ is a homomorphism from $$R$$ to $$S$$, or that $$\phi$$ is a homomorphism of rings

Crone Homomorphism Between Two by Two Matices with 0 in Bottom Left and Integers
Let $$R := \left\{ \begin{pmatrix} a & b \\ 0 & a \end{pmatrix} : a, b \in \mathbb{ Z } \right\}$$ along with the usual definition for matrix addition and matrix multiplication, and define $$\phi : R \to \mathbb{ Z }$$ be defined such that $\phi \left( \begin{pmatrix} a & b \\ 0 & a \end{pmatrix} \right) = a$ Show the following
• $$\phi$$ is a crone homomorphism
• Find $$\operatorname{ ker } \left( \phi \right)$$
• Show $$R / \operatorname{ ker } \left( \phi \right)$$ is ismorphic to $$\mathbb{ Z }$$
• is $$\operatorname{ ker } \left( \phi \right)$$ a prime ideal?

We start by showing $$\phi$$ is a crone homomorphism, note that we already know that $$R$$ and $$\mathbb{ Z }$$ are crones, now note the following: $\phi \left( \begin{pmatrix} a & b \\ 0 & a \end{pmatrix} \begin{pmatrix} c & d \\ 0 & c \end{pmatrix} \right) = \phi \left( \begin{pmatrix} ac & ac + bd \\ 0 & ac \end{pmatrix} \right) = ac = \phi \left( \begin{pmatrix} a & b \\ 0 & a \end{pmatrix} \right) \phi \left( \begin{pmatrix} c & d \\ 0 & c \end{pmatrix} \right)$ and that $\phi \left( \begin{pmatrix} a & b \\ 0 & a \end{pmatrix} + \begin{pmatrix} c & d \\ 0 & c \end{pmatrix} \right) = \phi \left( \begin{pmatrix} a + c & b + d \\ 0 & a + c \end{pmatrix} \right) = a + c = \phi \left( \begin{pmatrix} a & b \\ 0 & a \end{pmatrix} \right) + \phi \left( \begin{pmatrix} c & d \\ 0 & c \end{pmatrix} \right)$

We recognize that the $$2 \times 2$$ identity is the identity in this ring as well, therefore we confirm that $\phi \left( \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \right) = 1$ Thus $$\phi$$ is indeed a crone homomorphism

From the definition of $$\phi$$ we can conclude that $$\phi \left( \begin{pmatrix} a & b \\ 0 & a \end{pmatrix} \right) = 0$$ iff $$a = 0$$, therefore the kernel is all matrices of this form $\operatorname{ ker } \left( \phi \right) = \left\{ \begin{pmatrix} 0 & b \\ 0 & 0 \end{pmatrix}: b \in \mathbb{ Z } \right\}$

Recall the definition of $$R / \operatorname{ ker } \left( \phi \right)$$, it is \begin{align} \left\{ M + \operatorname{ ker } \left( \phi \right) : M \in R \right\} &= \left\{ \left\{ M + N : N \in \operatorname{ ker } \left( \phi \right) \right\} : M \in R \right\} \\ &= \left\{ \left\{ \begin{pmatrix} a & b \\ 0 & a \end{pmatrix} + \begin{pmatrix} 0 & c \\ 0 & 0 \end{pmatrix} : c \in \mathbb{ Z } \right\} : a, b \in \mathbb{ Z } \right\} \\ &= \left\{ \left\{ \begin{pmatrix} a & b + c \\ 0 & a \end{pmatrix} : c \in \mathbb{ Z } \right\} : a, b \in \mathbb{ Z } \right\} \\ & = \left\{ \left\{ \begin{pmatrix} a & d \\ 0 & a \end{pmatrix} : d \in \mathbb{ Z } \right\} : a \in \mathbb{ Z } \right\} \end{align} Let $$\operatorname{ tl } \left( M \right)$$ be defined as the top left coefficent in $$M$$, then for two matrices $$M, N$$ it should be clear that if $$\operatorname{ tl } \left( N \right) = \operatorname{ tl } \left( M \right)$$, then they generate the same coset, also that if $$\operatorname{ tl } \left( M \right) \neq \operatorname{ tl } \left( N \right)$$, then they generate different cosets. Also note that $$\operatorname{ tl } \left( M N \right) = \operatorname{ tl } \left( M \right) \cdot \operatorname{ tl } \left( N \right)$$ and that $$\operatorname{ tl } \left( M + N \right) = \operatorname{ tl } \left( M \right) + \operatorname{ tl } \left( N \right)$$. Finally given any coset, since we know that $$\operatorname{ tl } \left( X \right)$$ agrees for every $$X$$ in the coset, we can justify the construction of $$\overline{ \operatorname{ tl } } \left( N + \operatorname{ ker } \left( \phi \right) \right) = \operatorname{ tl } \left( N \right)$$

We claim that $$R / \operatorname{ ker } \left( \phi \right)$$ is an crone isomorphism to $$\mathbb{ Z }$$ using $$\overline{ \operatorname{ tl } }$$. We've actually already verified the that it respects addition and multiplication because of our discussion in the previous paragraph and also just note that if $$I$$ is the the identity then we know it's top left element is $$1 \in \mathbb{ Z }$$ so that $$\overline{ \operatorname{ tl } } \left( I \right) = \operatorname{ tl } \left( I \right) = 1$$ .

At this point we know that it's a homomorphism. So we just have to show it's a bijection to have an isomorphism so suppose that $$M + \operatorname{ ker } \left( \phi \right) \neq N + \operatorname{ ker } \left( \phi \right)$$ by the contrapositive of what we noted earlier since they generate different cosets, then $$\operatorname{ tl } \left( N \right) \neq \operatorname{ tl } \left( M \right)$$, therefoer $$\overline{ \operatorname{ tl } } \left( N + \operatorname{ ker } \left( \phi \right) \right) \neq \overline{ \operatorname{ tl } } \left( M + \operatorname{ ker } \left( \phi \right) \right)$$ , which shows that $$\operatorname{ tl }$$ is injective, it is clearly also surjective because since $$R$$ ranges over all matrices with integer coefficients, then if we pick a particular integer $$k \in \mathbb{ Z }$$ then there is always a matrix with that specific integer in it's top left component, say that matrix is $$X$$, then $$\overline{ \operatorname{ tl } } \left( X \right) = \operatorname{ tl } \left( X \right) = k$$ as needed, so $$\overline{ \operatorname{ tl } }$$ is an isomorphism.

Now we verify that $$\operatorname{ ker } \left( \phi \right)$$ is a prime ideal, so suppose that $$MN \in \operatorname{ ker } \left( \phi \right)$$, this means that $$\operatorname{ tl } \left( MN \right) = 0$$, but as noted earlier $$\operatorname{ tl } \left( M N \right) = \operatorname{ tl } \left( M \right) \operatorname{ tl } \left( N \right)$$, but we know that $$\mathbb{ Z }$$ so in this context we have $$\operatorname{ tl } \left( M \right) \operatorname{ tl } \left( N \right) = 0$$, but $$\mathbb{ Z }$$ is a domain therefore we must have wlog $$\operatorname{ tl } \left( M \right) = 0$$, but that implies that $$M \in \operatorname{ ker } \left( \phi \right)$$ which shows that $$\operatorname{ ker } \left( \phi \right)$$ is a prime ideal as neeeded.

Ring Isomorphism
Suppose that $$\phi$$ is a ring homomorphism and is a bijection, then we say that $$\phi$$ is a ring isomorphism
Homomorphisms carry Zero to Zero
Suppose that $$\phi : R \to S$$ is a homomorphism of rings, then $$\phi \left( 0 _ R \right) = 0 _ S$$