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Up to this moment I still have fuzzy ideas about some definitions in Abstract Algebra, especially about the difference between coset and ideal. Perhaps this question is a dumb one for you but I know precise definition plays a very crucial role in study of this math. Any plain English explanation would be much appreciated, especially any links to intuitive visualization of the ideas.

Thank you very much for your time and help.

A.Magnus
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  • See http://math.stackexchange.com/questions/23655/what-is-a-quotient-ring-and-cosets. – Dietrich Burde May 11 '14 at 12:58
  • How comfortable would you say that you are with the constructions of a quotient group and a quotient ring? – Carl Mummert May 11 '14 at 13:45
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    @CarlMummert - Very pathetically, I am not. I know G/N looks similar to congruence class module N, but that's about it. Thanks again for your time. – A.Magnus May 11 '14 at 19:41

1 Answers1

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There is not a direct analogy between a coset and an ideal - there's nothing "similar" to them.

There is a much closer analogy between an ideal in a ring and a normal subgroup in a group. A normal subgroup is a subgroup that has additional closure properties; not every subgroup is normal. An ideal in a ring is a subring that has extra closure properties; not every subring is an ideal.

These objects are important for quotient groups and quotient rings. Given a group $G$ and a subgroup $H$, you can form the quotient group $G/H$ if and only if $H$ is normal. Given a ring $R$ and a subring $S$, you can form the quotient ring $R/S$ if and only if $S$ is an ideal.

Carl Mummert
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  • Thanks for your early response. Could you please elaborate on what are those extra closure properties that make a subgroup normal, and those additional closure properties that make a subring an ideal? My understanding is that normal subgroup is one in which the left coset is the same as left coset. Thanks again for your response. – A.Magnus May 12 '14 at 13:04
  • Yes, for a subgroup $H$ to be normal, each left coset $xH$ must also be a right coset $Hy$ (maybe $y \not = x$). For a subring to be an ideal, the subring must be closed not only under multiplying two elements of the subring, but also closed under multiplying any element of the subring by any element of the original ring. So, although the idea of a subgroup or subring is very natural, it is not the idea that is needed to form quotient groups or quotient rings - more is needed, which is why these additional concepts of normal subgroup and ideal are necessary. – Carl Mummert May 12 '14 at 13:26
  • As always, thanks! – A.Magnus May 12 '14 at 18:39