The comma is the most variable punctuation sign and the most difficult one to pin down but there is a good, clear rule which says that that it is used in non-defining relative clauses. For example:
That painting, which is hanging on my wall, is by Picasso.
Here the commas mark a parenthesis. The sentence could even be written with brackets:
That painting (which is hanging on my wall) is by Picasso.
In A Guide to English Language Usage I summarise the use of relative pronouns in this way:
That is possible only in defining relative clauses; who and which can be used in all types.
That makes the situation clear for non-natives who are confused by the choice of relative pronouns in English. But there is a rule, or rather a pseudorule that is more common in the USA than in Britain, to the effect that which should not be used as the pronoun in defining relative clauses. It would insist that in this sentence:
The painting which/that I bought is by Picasso.
which is not a correct option. This is nonsense. There is a defining relative clause introduced by which in the first paragraph of this post.
Sentence first has an interesting piece about all this, taking as its text a sentence from the Guardian:
There is another lesson to the Petraeus affair. The former general fashioned for himself a role, which is much more significant than top generals have during wars.
The comma is clearly unnecessary and is actually misleading. It gets in the way of a straightforward understanding of the meaning.
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