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I saw the term "a ring with unity", and wikipedia mentions that unity refers to the identity element $1_R$ of the ring. So is a ring with unity the same as a ring with identity?

Thanks!

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Yes. ${}{}{}{}{}$ – Mariano Suárez-Alvarez♦ Apr 20 '11 at 2:46
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To add to Mariano's answer: the problem with "identity" is that it is sometimes unclear if "identity" refers to the additive identity, the identity map, or the multiplicative identity. That's why "with unity" or "with 1" is a common locution: it cuts down on possible misunderstandings. – Arturo Magidin Apr 20 '11 at 2:48
thanks for the clarifications! – mathcat Apr 20 '11 at 2:53

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Unity means a multiplicative identity, in the same way that things which are nth roots of 1 in the field $\mathbb{C}$ are called roots of "unity".

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