Prove that the ring \(M_2(\mathbb{R})\) contains a subring isomorphic to \(\mathbb{C}\).

Assume \(R\) is a commutative unital ring. Prove that the binomial theorem holds in \(R\).

Let \(R\) be a commutative ring. Prove that the set of nilpotent elements form an ideal (called the nilradical).

Prove that if \(R\) is a commutative ring and \(N\) is the nilradical, then \(R/N\) is reduced (has no non-zero nilpotents).

A commutative ring is called a local ring if it has a unique maximal ideal. Prove that if \(R\) is a local ring with maximal ideal \(M\) then every element of \(R \setminus M\) is a unit. Prove conversely that if \(R\) is a commutative unital ring where the set of nonunits forms an ideal \(M\), then \(R\) is a local ring with unique maximal ideal \(M\).

Prove that any subfield of \(\mathbb{R}\) must contain \(\mathbb{Q}\).

Let \(R\) be a non-trivial unital ring. An element \(e \in R\) is called an idempotent if \(e^2=e\). Assume \(e\) is an idempotent in \(R\) and \(er=re\) for all \(r \in R\). Prove that \(Re\) and \(R(1-e)\) are two-sided ideals of \(R\) and that \(R \cong Re \times R(1-e)\). Show that \(e\) and \((1-e)\) are identities for subrings \(Re\) and \(R(1-e)\) respectively.

Let \(R\) and \(S\) be unital rings. Prove that every ideal of \(R \times S\) is of the form \(I \times J\) where \(I\) is an ideal of \(R\) and \(J\) is an ideal of \(S\).