Just started learning algebra. So it's defined that ring is the ring not requiring a multiple 1, while unital ring does. Given a ring, is it always possible to extend it to a unital ring?
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4See here. – Pp.. Jan 21 '15 at 06:22
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1In many ways, some more suitable than the standard Dorroh adjunction (depending on context). – Bill Dubuque Jan 21 '15 at 14:59
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Starting with a ring $A$ having no multiplicative unit. Consider the direct sum $\mathbb{Z}\oplus A$ with the following addition $(m+\alpha)+(n+\beta)=(m+n)+(\alpha+\beta)$ and multiplication $(m+\alpha).(n+\beta)=mn +m\beta+\alpha n+\alpha\beta$ You can check it is a ring with unit $(1,0)$ and identifying $(0,\alpha)$ with $\alpha$ (and not $(0,\alpha)$ as in previous) makes it an extension of $A$
marwalix
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user209426
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It's a little confusing to use both ordered pairs and sums of pairs in your notation. It would probably be best to do what every book does and just use ordered pairs... – rschwieb Jan 21 '15 at 11:08