Cheyanne Leigh
2020-12-14
Answered

Let R and S be commutative rings. Prove that (a, b) is a zero-divisor
in $R\oplus S$ if and only if a or b is a zero-divisor or exactly one of a or
b is 0.

You can still ask an expert for help

Tuthornt

Answered 2020-12-15
Author has **107** answers

Let (a,b) is a zero-divisor in

Since (a,b) is zero-divisor, there exists (c,d) in

Since

Also,

Conversely:

Let a and b are zero divisors.

Since a is zero divisor,there exist element

Also, b is zero divisor, there exist element

Hence, for

By definition (a,b) is zero diisor in

Hence, proved

asked 2020-12-30

Let A := 5 1 16 -3 [2 3 9 4 (a) Find a basis for Nul A. (b) Find a basis for Col A.

asked 2022-05-02

$\frac{{\mathbb{F}}_{2}[X,Y]}{({Y}^{2}+Y+1,{X}^{2}+X+Y)}$ and $\frac{{\displaystyle \left({\mathbb{F}}_{2}\right[Y]}}{{\displaystyle ({Y}^{2}+Y+1)}}\frac{{\displaystyle \left)\right[X]}}{{\displaystyle ({X}^{2}+X+\overline{Y})}}$ are isomorphic

asked 2022-06-03

x=-3

asked 2021-09-15

Assume that T is a linear transformation. Find the standard matrix of T. $T:{\mathbb{R}}^{2}\to {\mathbb{R}}^{4},T\left({e}_{1}\right)=(3,1,3,1)\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}T\left({e}_{2}\right)=(-5,2,0,0),\text{}where\text{}{e}_{1}=(1,0)\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}{e}_{2}=(0,1)$ .

asked 2021-12-26

Assume that T is a linear transformation. Find the standard matrix of T.

T:${\mathbb{R}}^{2}\to {\mathbb{R}}^{4},\text{}T\left({e}_{1}\right)=(3,1,3,1),\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}T\left({e}_{2}\right)=(-5,6,0,0)$ , where ${e}_{1}=(1,0)$ and ${e}_{2}=(0,1)$

$A=$

T:

asked 2022-06-29

In an affine transformation $x\mapsto Ax+b$, $b$ represents the translation; but what does the matrix A represent exactly? In a 2D example, $A$ is a $2\times 2$ matrix, but what does each term represent?

asked 2021-02-25

Let U and W be vector spaces over a field K. Let V be the set of ordered pairs (u,w) where

(This space V is called the external direct product of U and W.)