Money was not invented.
Money is a natural phenomenon of human interaction - it is intrinsic to barter, it is that thing most commonly bartered for. Commerce cannot predate money by more than a handful of trades.
Sorry, the evidence does not support your assertion. summary, with a link to the book.
Barter came first, then debt, then money.
My forum avatar is a Sumerian word (amagi) meaning "freedom". Literally, the first sign (ama) means "mother" and the second sign (gi4) is the verb "to restore". Together, they are an idiom understood to mean freedom. Sumerians had a complex economic system centered around their temples. Writing itself was invented in the temples of Sumer to facilitate exactly this accounting of debts. People would get themselves into debt and end up essentially selling themselves into slavery to settle up. Over time, this turned out to be a problem, because people don't like being slaves, so if there were too many, they had a tendency to revolt and wreck up the place. So, the jubilee was invented, where debts were wiped out, slaves set free (and able to go home to their mothers, hence the idiom), etc, every few years or so.
This was all well established at least 2000 years before money was invented. And for all we know, even thousands of years earlier (it is really hard to pin down intangible concepts like debt prior to the invention of writing, but it is very clear that writing was invented for accounting).