# Thread: Proving Rings (sorry for the Tex error)

1. ## Proving Rings (sorry for the Tex error)

$
Prove that \left<\frac{R}{I}\,,+\,,\cdot\right> is a ring.
$

2. Originally Posted by DanielThrice
$
Prove that \left<\frac{R}{I}\,,+\,,\cdot\right> is a ring.
$

Perhaps you mean to prove that if $R$ is a conmutative ring, $I\subset R$ ideal of $R$, then $R/I$ is a commutative ring with the standard operations $(x+I)+(y+I)=(x+y)+I$ and $(x+I)(y+I)=xy+I$ . If so, start proving that the sum is well defined that is, if $x+I=x'+I$ and $y+I=y'+I$ then, $(x+I)+(y+I)=(x'+I)+(y'+I)$ .

Fernando Revilla

3. Sorry, (I was half asleep last night) here is what we already know about the problem:

Let R = < R, +, *> (* means multiplication) be a ring, and let I < R be an additive subgroup of < R, + >.
Consider the set of cosets

R/I = {a + I: a is an element of R}

equipped with its own operations + and * defined by

(a + I) + (b + I) = (a + b) + I
(a + I) (b + I) = ab + I

Assume that the operations + and are well-defined <---> the
additive subgroup I satises the following conditions:

ab is an element of I for all a in R and b in I

ba is an element of I for all a in R and b in I
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 11, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9675456881523132, "perplexity": 572.6248108843699}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187820927.48/warc/CC-MAIN-20171017052945-20171017072945-00298.warc.gz"}