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- Prove that the intersection of two ideals of a ring is an ideal.

If*I*and*J*are ideals of a commutative ring with identity, let*IJ*be the ideal generated by the set {*ij*|*i*∈*I*,*j*∈*J*}.

Prove that*IJ*⊆*I*∩*J*.

The ideal*I*+*J*is the ideal generated by*I*∪*J*.

Let*I*be the ideal < 4 > in**Z**and let*J*= < 6 > . Describe the ideals*IJ*,*I*∩*J*and*I*+*J*. - Prove that the ring
**Z**_{n}is a principal ideal domain for any*n*. How would you determine the number of ideals of**Z**_{n}? - Prove that 2
**Z**and 3**Z**are isomorphic as abelian groups but not as rings (under, of course, the usual addition and multiplication).

The addititive groups**Z**_{6}and**Z**_{2}×**Z**_{3}are isomorphic as groups. Show that they are also isomorphic as rings.

[Hint: A suitable group isomorphism maps 1 ∈**Z**_{6}to (1, 1) ∈**Z**_{2}×**Z**_{3}.]

More generally, show that if*m*,*n*are coprime integers then**Z**_{mn}and**Z**_{m}×**Z**_{n}are isomorphic both as groups and as rings. - Prove that the map from
**Z**_{12}to**Z**_{4}given by*n*↦*n**mod*4 for*n*∈**Z**_{12}is a ring homomorphism. What is its kernel?

Is the map from**Z**_{14}to**Z**_{4}given by*n*↦*n**mod*4 for*n*∈**Z**_{14}a ring homomorphism? - Which of the maps from
**C**to**C**given by the following are ring homomorphisms.

*x*+*yi*↦*x**x*+*yi*↦*x*-*yi**x*+*yi*↦ |*x*+*iy*|*x*+*yi*↦*y*+*xi* - Show that the units (multiplicatively invertible elements) in any ring form a group under multiplication.

Denote the group of units of**Z**_{n}by*U*_{n}.

For various values of*n*(as far as you can!) identify the groups*U*_{n}as products of cyclic groups. Can you spot any pattern? Question 3 above should be helpful. - Describe all the ring homomorphisms from
**Z**_{12}onto**Z**_{4}.

Describe all the ring homomorphisms from**Z**_{12}to**Z**_{5}.

For which values of*m*,*n*can one find a ring homomorphism from**Z**_{m}*onto***Z**_{n}? - Look at possible addition and multiplication tables and prove that up to isomorphism there are two rings of order 2.

If the additive group of a ring is cyclic, generated (say) by an element*a*, prove that the multiplication is determined once you know*a*.*a*. Hence determine how many different rings of order 3 there are.Prove that there are three non-isomorphic rings of order 4 whose additive group is cyclic.

If you have sufficient determination you can try and establish how many non-isomorphic rings there are whose additive group is the Klein 4-group

**Z**_{2}×**Z**_{2}.

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