# Michigan-State-University-Abstract-Algebra-eye-catch

• Difference Between Ring Homomorphisms and Module Homomorphisms Let $R$ be a ring with $1$ and consider $R$ as a module over itself. (a) Determine whether every module homomorphism $\phi:R\to R$ is a ring homomorphism. (b) Determine whether every ring homomorphism $\phi: R\to R$ is a module homomorphism. (c) If $\phi:R\to R$ is both a […]
• Condition that Two Matrices are Row Equivalent We say that two $m\times n$ matrices are row equivalent if one can be obtained from the other by a sequence of elementary row operations. Let $A$ and $I$ be $2\times 2$ matrices defined as follows. $A=\begin{bmatrix} 1 & b\\ c& d \end{bmatrix}, \qquad […] • Subspaces of Symmetric, Skew-Symmetric Matrices Let V be the vector space over \R consisting of all n\times n real matrices for some fixed integer n. Prove or disprove that the following subsets of V are subspaces of V. (a) The set S consisting of all n\times n symmetric matrices. (b) The set T consisting of […] • Subset of Vectors Perpendicular to Two Vectors is a Subspace Let \mathbf{a} and \mathbf{b} be fixed vectors in \R^3, and let W be the subset of \R^3 defined by \[W=\{\mathbf{x}\in \R^3 \mid \mathbf{a}^{\trans} \mathbf{x}=0 \text{ and } \mathbf{b}^{\trans} \mathbf{x}=0\}.$ Prove that the subset $W$ is a subspace of […]
• Find a Basis for the Range of a Linear Transformation of Vector Spaces of Matrices Let $V$ denote the vector space of $2 \times 2$ matrices, and $W$ the vector space of $3 \times 2$ matrices. Define the linear transformation $T : V \rightarrow W$ by \[T \left( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \right) = \begin{bmatrix} a+b & 2d \\ 2b - d & -3c \\ 2b - c […]
• Prime Ideal is Irreducible in a Commutative Ring Let $R$ be a commutative ring. An ideal $I$ of $R$ is said to be irreducible if it cannot be written as an intersection of two ideals of $R$ which are strictly larger than $I$. Prove that if $\frakp$ is a prime ideal of the commutative ring $R$, then $\frakp$ is […]
• All the Eigenvectors of a Matrix Are Eigenvectors of Another Matrix Let $A$ and $B$ be an $n \times n$ matrices. Suppose that all the eigenvalues of $A$ are distinct and the matrices $A$ and $B$ commute, that is $AB=BA$. Then prove that each eigenvector of $A$ is an eigenvector of $B$. (It could be that each eigenvector is an eigenvector for […]
• The Index of the Center of a Non-Abelian $p$-Group is Divisible by $p^2$ Let $p$ be a prime number. Let $G$ be a non-abelian $p$-group. Show that the index of the center of $G$ is divisible by $p^2$. Proof. Suppose the order of the group $G$ is $p^a$, for some $a \in \Z$. Let $Z(G)$ be the center of $G$. Since $Z(G)$ is a subgroup of $G$, the order […]