Is $\{a+bi: a,b \in \mathbb{Z} \} $ and $i^2=-1$, a ring under the usual operations of addition and multiplication? It needs to be closed under addition and multiplication which it seems like Addition needs to be commutative and associative which is quite straightforward. It must have an identity element which I am not sure about and it must have an additive inverse which I am also unclear about Multiplication must be associative which it should be and then it must be distributive over addition meaning that a*(b+c)=a*b+a*c which i also think it is.

  • $\begingroup$ Are you assuming that $\Bbb C$ is already known, and the given set is a subset of $\Bbb C$, or are you instead viewing it as a set of formal expressions? $\endgroup$ – Bill Dubuque Aug 19 '12 at 7:40
  • $\begingroup$ I am assuming that we are viewing it as a set of formal expressions $\endgroup$ – math101 Aug 19 '12 at 13:55

The multiplicative identity is $1+0i$, the additive identity is $0+0i.$

Associativity (and commutativity) of addition and multiplication follow from the same properties of complex numbers in general (which you should verify if you haven't done so before!), as does distributivity.

You can easily check that it is multiplicatively closed by expanding $(a+bi)(c+di)$ for $a,b,c,d\in\mathbb Z.$

The additive inverse of $a+bi$ is $-a-bi.$

  • 1
    $\begingroup$ Thanks I really appreciate you help $\endgroup$ – math101 Aug 19 '12 at 3:17

Yes it is. This ring is known as the Gaussian integers, and you can learn a great deal about it on Wikipedia.

  • $\begingroup$ Thanks I'll take a look at it :) $\endgroup$ – math101 Aug 19 '12 at 3:17

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