I can not understand key difference between them, because all on the web uses mathematical notations...Can anyone explain in easy way?...please.Thanks in advance.
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In a group, you can multiply element and divide (example : non zero numbers). In general though, you don't ask that $a \times b = b \times a$ : you can come up with abstract examples of groups where it's not true).
In a ring, you can add, substract, multiply but not necessarily divide. Example : integers. You can add multiply and substract integers and it will remain integers, but if you try to divide $3$ by $2$ you will leave the world of integers.
In a field, you can do everything you can in a ring but additionnally you can always divide by non zero elements. Example : the rational numbers, the real numbers, the complex numbers.
Hal
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Think of the standard examples:
A ring is "like" the integers $\mathbb Z$. You can add and subtract any two elements, but you can not necessarily divide them by each other. For example, the element $\frac 12$ is not an integer, even though both $1$ and $2$ are.
A field is "like" the fractions $\mathbb Q$. You can add and substract, and you can also divide them. That is, if $p$ and $q$ are two rational numbers, then their quotient $p/q$ is also a rational number.
A group is something where you can multiply things together, but you cannot necessarily add and subtract them. So for example, if you forget about addition, then $\mathbb{Q} \backslash \{ 0\}$ is a group. The product of any two rational numbers is a rational number. You must also be able to find "inverses", that is, to divide by elements. That is why the integers is not a group under multiplication.
Fredrik Meyer
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1Probably you meant $\mathbb Q^\times$ in your example of "group". – paul garrett May 09 '14 at 13:58
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@paulgarrett Thank you! – Fredrik Meyer May 09 '14 at 15:19