I don't fully understand the terms "natural homomorphism" or "homomorphic image." It says that if S is a homomorphic image of R then R and S might have the same algebraic properties.
It seems that this section is important because it relates things that can be helpful when writing proofs. It is interesting that the section says, "A concept developed with one idea in mind may have unexpected linkages with other important mathematical concepts. That is precisely the situation here." It's neat that there are other results than just what they had wanted.
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