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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

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    Define $x \star y = x+y-1$. Now the identity element is $1$. – CyclotomicField Jan 17 '24 at 21:29
  • Generally rings don't contain $\Bbb Z$ so it makes no sense to compare their elements to integers. But if $R$ has a multiplicative neutral element $1_R$ then we get an image of $\Bbb Z$ in $R,$ via the map $,h,:, 1_\Bbb Z\mapsto 1_R,,$ namely $\Bbb Z/n,,$ for $, (n) =\ker h= $ characteristic of $R$. When $,n=0,$ the image of $\Bbb Z$ is an embedding so we often denote the integer images using integer notation, even though they may be polynomials, power series, matrices, functions, hypercomplex numbers etc. When $n>0$ we get images of modular integers from $\Bbb Z_n = \Bbb Z/n.$ – Bill Dubuque Jan 17 '24 at 22:42
  • For example we often abuse notation by writing $1$ for the unit element of $,\Bbb Z_n[x],$ when technically we should $,[1]_n x^0,$ or $,([1]_n, [0]_n, [0]_n,\ldots),,$ depending on how we define polynomial and quotient rings. But it is not the notation (syntax) that matters, rather it is the meaning (semantics), i.e. how that element relates to the others under the operations, here being that it satisfies the definition of a multiplicative neutral element (which can be verified by examining the operation tables of the structure). – Bill Dubuque Jan 17 '24 at 22:54
  • Generally in abstract algebra we only care about structure up to isomorphism, i.e. structures with the same arithmetic (operation tables, up to order) are considered the same (cf. the Remark here). If a ring contains a subring isomorphic to some known ring, them we may reuse that known rings notation in order to aid our intuition in comprehending the larger ring. This relationship is abstracted by the notion of an associative algebra, a ring containing a central image of the known ring. – Bill Dubuque Jan 17 '24 at 23:08
  • In the boolean ring $P(S)$ of subsets of a fixed set $S$ the zero element is the empty set, addition is the symmetric difference, and multiplication is the intersection. But we can still use $0,1,+,\cdot$ notation for these. – Jyrki Lahtonen Jan 18 '24 at 08:18
  • @JyrkiLahtonen yes, I know it, but my question about whether it is valid for fields as like rings. –  Jan 18 '24 at 13:23
  • @JyrkiLahtonen do you know examples for fields as you give example for rings, i liked it very much –  Jan 18 '24 at 13:24

2 Answers2

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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 $R$.

Gerry Myerson
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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?

Gerry Myerson
  • 179,216