\documentclass{article}
% Change "article" to "report" to get rid of page number on title page
\usepackage{amsmath,amsfonts,amsthm,amssymb}
\usepackage{setspace}
\usepackage{fancyhdr}
\usepackage{lastpage}
\usepackage{extramarks}
\usepackage{chngpage}
\usepackage{soul,color}
\usepackage{graphicx,float,wrapfig,tikz}

% In case you need to adjust margins:
\topmargin=-0.45in      %
\evensidemargin=0in     %
\oddsidemargin=0in      %
\textwidth=6.5in        %
\textheight=9.0in       %
\headsep=0.25in         %


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}
\begin{spacing}{1.1}
\newpage

$$ \text{\Large \textbf{Math 81/111 Homework \#1}}
$$

$$
\text{Due January 20, 2016}
$$
\\
\hspace{-0.2cm}
\textbf{1.}
	Recall that a subring $S$ of a ring $R$ is an additive subgroup of $R$ which is closed 
	under multiplication and contains the multiplicative identity element. \\
	(a) Determine all subrings of $\mathbb{Z}$.\\
	(b) Determine all subrings of $\mathbb{Z} \times \mathbb{Z}$.\\
	(c) Determine all ideals of $\mathbb{Z} \times \ldots \times \mathbb{Z}$.\\
	\\
\textbf{2.}
	Give an example of a commutative ring $R$ and two distinct ideals $I$ and $J$ of $R$ 
	such that $I \cap J \neq IJ$.\\
	\\
\textbf{3.}
	Let $R$ be a commutative ring, $r \in R$. Prove that there is a unique ring 
	homomorphism $f:\mathbb{Z}[X] \to R$ such that $f(X) = r$. Show that the 
	image of $f$ is the smallest subring of $R$ that contains $r$.\\
	\\ 
\textbf{4.}
	(a) An element $a$ of a ring $R$ is called $\emph{nilpotent}$ if $a^n=0$ for some
	 $n \in \mathbb{N}$. Show that if $R$ is commutative, then the set $Nil(R)$ of all 
	 nilpotent elements in $R$ is an ideal (the $\emph{nilradical}$ of $R$). Prove that 
	 the quotient ring $R/Nil(R)$ has no nonzero nilpotent elements.\\
	(b) Prove that a polynomial $f(X) = a_0+a_1X+\ldots+a_nX^n \in R[X]$ ($R$ is a 
	commutative ring) is nilpotent if and only if all $a_i$ are nilpotent in $R$.\\
	\\
\textbf{5.}
	(a) Prove that if $a$ is a nilpotent element of a ring $R$, then the element $1+a$ is invertible. \\
	(b) Prove that a polynomial $f(X) = a_0+a_1X+\ldots+a_nX^n \in R[X]$ ($R$ is a 
	commutative ring) is invertible if and only if $a_0$ is invertible and $a_i$ is nilpotent 
	in $R$ for all $i=1,\ldots,n$.\\
	\\
		
\end{spacing}
\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%