definition of the operations and additive identity for fields

Question

I know that if the ordered triple $(S,+,*)$ is a ring , then the sign $+$ and $*$ can represent some operations different from their original usage such as addition and multiplication(for example boolean ring). Moreover, ring zero ,i.e the identity element of operation $+$, can be different from integer zero if we do not use symbol $+$ as addition.

In the definition of field, if $[A,+.*]$ is a field on set A ,then

• $[A,+,*]$ is a ring

• $(A-\{0\},*)$ is abelian group

I want to ask something here. I have never encounter with such fields whose ring zero is not different from integer zero. In every fields such as the sets are complex numbers,real numbers etc, the zero in the notation of $(A-\{0\},*)$ always represent integer zero,and addition and multiplication symbol always used in their usual way (just basic algebra usage)

So, is there any other example such that the zero in $(A-\{0\},*)$ is different from integer zero. Moreover, the symbols addition and multiplication can represent any other binary operations in fields (like in rings such as boolean rings), or do we strictly have to use ordinary addition and multiplication operations as binary operations for fields. I am looking for answer of these two questions .

I am sorry if my question does not fit this site,because I am beginner in both algebra and this site. I read books and ask questions in school, but I am embarrassed to ask this type of basic questions to my teacher

Answer 1

Let $F$ be a field, let $G$ be any old set with the same cardinality as $F$, let $\phi:F\to G$ be one-one and onto. Define addition on $G$ by $g+h=\phi(\phi^{-1}(g)+\phi^{-1}(h))$, multiplication by $gh=\phi(\phi^{-1}(g)\phi^{-1}(h))$. Then $G$ is a field, with additive identity element $\phi(z)$, where $z$ is the additive identity element of $F$. So, the additive identity element of $G$ could be $17$, or George, or whatever you choose.

Less artificial: let $R$ be an integral domain, let $M$ be a maximal ideal of $R$, then the set of cosets of $M$ in $R$ is a field under the usual definitions of coset addition and multiplication, and its additive identity is the coset of $M$ containing the additive identity of $R$.

Answer 2

Let $A=\begin{pmatrix} 0&2\\ 1&0 \end{pmatrix}$ and let $I$ be the $2\times2$ identity matrix. Now consider the subset $K=\{xI+yA\vert x,y\in\mathbb{Q}\}\subset\mathbb{Q}^{2\times 2}$ of the $2\times 2$ matrices with rational coefficients. Now $K$ is a field and its zero element is the zero matrix, the addition is the matrix addition and the multiplication is the matrix multiplication. Does this fit your criterion?