You don't have to do an inflation adjustment on an instantaneous earnings-related rate. A firm with 10% ROE is likely to have 10% ROE the next year with smaller dollars. With a shrunken yardstick you get bigger R and proportionally bigger E as well: the same ratio.

I don't think this is necessarily true.
E could increase much more slowly (proportionally) due to inflation than R.
Let's take an example.
Beginning assets = $100 monetary, $200 non-monetary (book value with no depreciation).
Liabilities = 0.
Beginning equity = $300.
No inflation.
Earnings = $100.
Ending equity = $400 (all earnings go to retained earnings.)
If we calculate ROE on the average equity, it comes to 2/7 ($100/$350) = 28.57%. On the beginning equity, it is $100/$300 = 1/3.

Now assume 10% inflation (not hyperinflation which will trigger asset re-evaluation and change in equity.)
Beginning equity = $300.
Earnings = $110.
Ending equity = $410.
Average equity = $355.
ROE = 110/355 = 30.985% ~ 31%. On beginning equity, ROE = $110/$300 > 1/3.

If you have to restate assets to $100 (monetary) + $220 (non-monetary) = $320 then your ending equity will be $320 + $110 = $430.
Average = $365. ROE = 30.1369%.

Conclusion:
IF all net assets (assets - liabilities) are non-monetary AND you calculate ROE on beginning equity AND there is no depreciation, THEN your statement will hold true.

In our case, assume $300 assets = $300 equity at the beginning, and no cash, and no depreciation charges; then at the end both would be valued at $330, not counting the recently added $110 in earnings, and ROE will remain at 1/3 = $110/$330.

Otherwise, ROE will increase with inflation.