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1

Define the two rings \({\mathbb Z}_{11}\) and \({\mathbb Z}_{12}\) with the commands `R = Integers(11)`

and `S = Integers(12)`

. For each ring, use the relevant command to determine: if the ring is finite, if it is commutative, if it is an integral domain and if it is a field. Then use single Sage commands to find the order of the ring, list the elements, and output the multiplicative identity (i.e. \(1\text{,}\) if it exists).

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2

Define `R`

to be the ring of integers, \({\mathbb Z}\text{,}\) by executing `R = ZZ`

or `R = Integers()`

. A command like `R.ideal(4)`

will create the principal ideal \(\langle 4\rangle\text{.}\) The same command can accept more than one generator, so for example, `R.ideal(3, 5)`

will create the ideal \(\{a\cdot 3+ b\cdot 5\mid a,b\in{\mathbb Z}\}\text{.}\) Create several ideals of \({\mathbb Z}\) with two generators and ask Sage to print each as you create it. Explain what you observe and then create code that will test your observation for thousands of different examples.