It might not be grammatically wrong to say "I have no peach", but it is not a construction I have ever heard in normal speech (in British English). "I have no peaches" would be more likely.
And to describe people who would use "I ain't got no peach" as uneducated or using an ignorant vernacular is somewhat perjorative: "ain't" may not be standard English and shouldn't be used in formal contexts, but it is perfectly normal in many dialects and informal settings in both Britain and North America (and possibly also in Australia and New Zealand).
I say things like "I have no peach" all the time; seems a more poignant statement to me. In the first couple hundred results in the Corpus of Contemporary American English, many things following "I have no" are somewhat more abstract (although countable): choice, idea, opinion, clue, excuse. However, there's also "I have no fixed routine"; "I have no stable"; "I have no brother"; and "I have no country."
"ain't got no peach" is a double negative. If you don't have no peach, then you have a peach or peaches. Otherwise, you have no peach.
I wrote "...people who choose to speak in an ignorant vernacular." I wasn't calling the people who speak this way ignorant, since they speak this way by choice.
If you can read in ukrainian: https://uk.wikipedia.org/wiki/%D0%A0%D0%BE%D0%B4%D0%BE%D0%B2%D0%B8%D0%B9_%D0%B2%D1%96%D0%B4%D0%BC%D1%96%D0%BD%D0%BE%D0%BA_%D0%BE%D0%B4%D0%BD%D0%B8%D0%BD%D0%B8_%D1%96%D0%BC%D0%B5%D0%BD%D0%BD%D0%B8%D0%BA%D1%96%D0%B2_%D1%87%D0%BE%D0%BB%D0%BE%D0%B2%D1%96%D1%87%D0%BE%D0%B3%D0%BE_%D1%80%D0%BE%D0%B4%D1%83_%D0%B4%D1%80%D1%83%D0%B3%D0%BE%D1%97_%D0%B2%D1%96%D0%B4%D0%BC%D1%96%D0%BD%D0%B8
The trouble is that having used Google translate, it also translates the words that you want left in Ukrainian as examples etc. I'm sure somewhere there must be an English language explanation ... the only one I've found so far is here: http://www.ukrainianlanguage.org.uk/read/unit07/page7-3.htm
"Whilst most masculine nouns in the genitive end in а or я, there are certain categories of nouns (e.g. abstract concepts, weights and measures, mass nous, locations and institutions, emotions and states of mind, natural phenomena etc.) which mostly have an ending in у or ю, e.g. університет – університету, дощ – дощу, ідеал – ідеалу etc.
In some nouns a change in genitive ending also denotes a change of meaning, e.g. рака – (of a) lobster, раку – (of) cancer, соняшника – (of) sunflower, соняшнику – (of) sunflower seed. You need to be aware of this point, but it is not covered in detail in this course."
Is "I don't have the peach" wrong?
I know the course doesn't accept things like "I have the peach" for "У мене є персик" (about the only time it doesn't accept both kinds of articles indiscriminately) and I figure that's because the sentence is about existence of any peach rather than location of a specific peach. But I wouldn't have thought there'd be a similar distinction for negative statements.
But does Ukrainian have a way to go about pithily specifying a definite peach that I don't happen to have at the moment (like maybe it fell out of the picnic basket in my trunk without my noticing and now I've lugged the basket quite a ways for the picnic)?
Would that have to be "Я не маю персика?"
Forgive me, but I don't know what genitive means. But i suppose you're referring to the form of words when you state the absence of the word in question.
Okay then, in most cases I've encountered on this course at least, the genitive case and plural of most words end in an "и". Немає води, немає книги Are these not the same form as the plural?
At least words ending in "а" in their original form, whereas words ending in a noun would then add an "а".
I'm not saying all words are the same as their plural form. But enough for me to make the mistake I mentioned. I suppose I am looking for the easiest way to memorize these different forms of words, and comparing the words when referring an absence of it, to words in plural form, seemed to work....for a short time :)