Basic Exercise Problems in Module Theory
Let $R$ be a ring with $1$ and $M$ be a left $R$-module.
(a) Prove that $0_Rm=0_M$ for all $m \in M$.
Here $0_R$ is the zero element in the ring $R$ and $0_M$ is the zero element in the module $M$, that is, the identity element of the additive group $M$.
To simplify the […]
Welcome to Problems in Mathematics
Welcome to my website.
I post problems and their solutions/proofs in mathematics.
Most of the problems are undergraduate level mathematics.
Here are several topics I cover on this website.
Topics
Linear Algebra
Group Theory
Ring Theory
Field Theory, Galois Theory
Module […]
Sylow’s Theorem (Summary) In this post we review Sylow's theorem and as an example we solve the following problem.
Show that a group of order $200$ has a normal Sylow $5$-subgroup.
Review of Sylow's Theorem
One of the important theorems in group theory is Sylow's theorem.
Sylow's theorem is a […]
The Center of a p-Group is Not Trivial
Let $G$ be a group of order $|G|=p^n$ for some $n \in \N$.
(Such a group is called a $p$-group.)
Show that the center $Z(G)$ of the group $G$ is not trivial.
Hint.
Use the class equation.
Proof.
If $G=Z(G)$, then the statement is true. So suppose that $G\neq […]
Independent Events of Playing Cards
A card is chosen randomly from a deck of the standard 52 playing cards.
Let $E$ be the event that the selected card is a king and let $F$ be the event that it is a heart.
Prove or disprove that the events $E$ and $F$ are independent.
Definition of Independence
Events […]
Mathematics About the Number 2018 Happy New Year 2018!!
Here are several mathematical facts about the number 2018.
Is 2018 a Prime Number?
The number 2018 is an even number, so in particular 2018 is not a prime number.
The prime factorization of 2018 is
\[2018=2\cdot 1009.\]
Here $2$ and $1009$ are […]