A ring is Local if and only if the set of Non-Units is an Ideal

Problem 526

A ring is called local if it has a unique maximal ideal.

(a) Prove that a ring $R$ with $1$ is local if and only if the set of non-unit elements of $R$ is an ideal of $R$.

(b) Let $R$ be a ring with $1$ and suppose that $M$ is a maximal ideal of $R$.
Prove that if every element of $1+M$ is a unit, then $R$ is a local ring.

 

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The Quotient Ring by an Ideal of a Ring of Some Matrices is Isomorphic to $\Q$.

Problem 525

Let
\[R=\left\{\, \begin{bmatrix}
a & b\\
0& a
\end{bmatrix} \quad \middle | \quad a, b\in \Q \,\right\}.\] Then the usual matrix addition and multiplication make $R$ an ring.

Let
\[J=\left\{\, \begin{bmatrix}
0 & b\\
0& 0
\end{bmatrix} \quad \middle | \quad b \in \Q \,\right\}\] be a subset of the ring $R$.

(a) Prove that the subset $J$ is an ideal of the ring $R$.

(b) Prove that the quotient ring $R/J$ is isomorphic to $\Q$.

 

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Is the Given Subset of The Ring of Integer Matrices an Ideal?

Problem 524

Let $R$ be the ring of all $2\times 2$ matrices with integer coefficients:
\[R=\left\{\, \begin{bmatrix}
a & b\\
c& d
\end{bmatrix} \quad \middle| \quad a, b, c, d\in \Z \,\right\}.\]

Let $S$ be the subset of $R$ given by
\[S=\left\{\, \begin{bmatrix}
s & 0\\
0& s
\end{bmatrix} \quad \middle | \quad s\in \Z \,\right\}.\]

(a) True or False: $S$ is a subring of $R$.

(b) True or False: $S$ is an ideal of $R$.

 

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Top 10 Popular Math Problems in 2016-2017

It’s been a year since I started this math blog!!

More than 500 problems were posted during a year (July 19th 2016-July 19th 2017).

I made a list of the 10 math problems on this blog that have the most views.

Can you solve all of them?


The level of difficulty among the top 10 problems.
【★★★】 Difficult (Final Exam Level)
【★★☆】 Standard(Midterm Exam Level)
【★☆☆】 Easy (Homework Level)
 

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How to Prove a Matrix is Nonsingular in 10 Seconds

Problem 509

Using the numbers appearing in
\[\pi=3.1415926535897932384626433832795028841971693993751058209749\dots\] we construct the matrix \[A=\begin{bmatrix}
3 & 14 &1592& 65358\\
97932& 38462643& 38& 32\\
7950& 2& 8841& 9716\\
939937510& 5820& 974& 9
\end{bmatrix}.\]

Prove that the matrix $A$ is nonsingular.

 

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